Where does one even need Bernoulli's Inequality?












10












$begingroup$


Which Theorems/Lemmas/Results actually use Bernoulli's inequality? I don't seem to remember using it very often - which probably makes sense, as it's not a very strong inequality and can be proven easily.



However, where do you actually use Bernoulli?










share|cite|improve this question











$endgroup$












  • $begingroup$
    The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 16:17










  • $begingroup$
    Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
    $endgroup$
    – Kezer
    Dec 4 '18 at 16:42
















10












$begingroup$


Which Theorems/Lemmas/Results actually use Bernoulli's inequality? I don't seem to remember using it very often - which probably makes sense, as it's not a very strong inequality and can be proven easily.



However, where do you actually use Bernoulli?










share|cite|improve this question











$endgroup$












  • $begingroup$
    The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 16:17










  • $begingroup$
    Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
    $endgroup$
    – Kezer
    Dec 4 '18 at 16:42














10












10








10


6



$begingroup$


Which Theorems/Lemmas/Results actually use Bernoulli's inequality? I don't seem to remember using it very often - which probably makes sense, as it's not a very strong inequality and can be proven easily.



However, where do you actually use Bernoulli?










share|cite|improve this question











$endgroup$




Which Theorems/Lemmas/Results actually use Bernoulli's inequality? I don't seem to remember using it very often - which probably makes sense, as it's not a very strong inequality and can be proven easily.



However, where do you actually use Bernoulli?







real-analysis inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 4 '18 at 16:14









José Carlos Santos

155k22124227




155k22124227










asked Dec 4 '18 at 16:00









KezerKezer

1,313421




1,313421












  • $begingroup$
    The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 16:17










  • $begingroup$
    Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
    $endgroup$
    – Kezer
    Dec 4 '18 at 16:42


















  • $begingroup$
    The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 16:17










  • $begingroup$
    Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
    $endgroup$
    – Kezer
    Dec 4 '18 at 16:42
















$begingroup$
The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
$endgroup$
– Robert Wolfe
Dec 4 '18 at 16:17




$begingroup$
The Triangle Inequality, Sums of Squares, and Bernoulli's can bootstrap basically all relevant inequalities in analysis.
$endgroup$
– Robert Wolfe
Dec 4 '18 at 16:17












$begingroup$
Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
$endgroup$
– Kezer
Dec 4 '18 at 16:42




$begingroup$
Which are „all relevant inequalities in analysis“ to you (and where does one use Bernoulli)?
$endgroup$
– Kezer
Dec 4 '18 at 16:42










5 Answers
5






active

oldest

votes


















14












$begingroup$

Bernoulli's Inequality can prove the AM-GM Inequality. From this fact, you could derive Young's Inequality, Holder's Inequality, Minkowski's Inequality, and in turn any that follow from those.



Let $a_1, a_2, ldots, a_n$ be $n$ positive real numbers. Let us define
$$A_k=frac{a_1+a_2+cdots+a_k}{k}$$
for every $1leq kleq n$. Bernoulli's Inequality in the form $x^kgeq 1+k(x-1)$ then implies
$$left(frac{A_k}{A_{k-1}}right)^kgeq 1+kleft(frac{A_k}{A_{k-1}}-1right)$$
which after some algebraic hyjinx results in
$$A_k^kgeq a_kA_{k-1}^{k-1},.$$
This in turn implies
$$A_n^ngeq a_nA_{n-1}^{n-1}geq a_na_{n-1}A_{n-2}^{n-2}geq cdotsgeq a_ncdots a_2a_1$$
which gives
$$sqrt[n]{a_1cdots a_n}leqfrac{a_1+a_2+cdots+a_n}{n},.$$
Intuitively, what's happening here is that we can order the values $A_1, A_2, ldots, A_n$ so that the subsequent quotients $A_k/A_{k-1}$ are close to $1$, which is where Bernoulli's is precise.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
    $endgroup$
    – Kezer
    Dec 4 '18 at 17:25






  • 4




    $begingroup$
    @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 17:47










  • $begingroup$
    Excellent intuition, thank you!
    $endgroup$
    – Kezer
    Dec 4 '18 at 19:14










  • $begingroup$
    Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 20:42






  • 1




    $begingroup$
    @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
    $endgroup$
    – Robert Wolfe
    Dec 5 '18 at 18:08



















8












$begingroup$

When dealing with asymptotics of probabilities, the expression $(1-p)^n$ comes up all the time. The most convenient way to handle it is with the inequalities $$1 - pn le (1-p)^n le e^{-pn}$$ where the lower bound is Bernoulli's inequality. (I'm assuming here that $p in [0,1]$ and $n$ is a natural number.) Actually, as mentioned in p4sch's answer, the upper bound is also a consequence of Bernoulli's inequality, via the inequality $1 + x le e^{x}$.



For example, the result that monotone properties of the Erdős–Rényi model of random graphs have thresholds relies on the fact that if you take the union of $k$ copies of $mathcal G_{n,p}$, the graph you get (which has the distribution $mathcal G_{n,1-(1-p)^k}$) can be thought of as a subgraph of $mathcal G_{n,kp}$. This implies that as the edge probability $p$ scales linearly, the probability that your graph lacks a monotone property decays exponentially: $$Pr[mathcal G_{n,kp} text{ lacks property $M$}] le Pr[mathcal G_{n,p} text{ lacks property $M$}]^k.$$ For more details, see Theorem 1.7 in this textbook.



Many inequalities can prove each other and it's hard to say you ever "need" a particular result. This, however, is an example where Bernoulli's inequality is the most convenient tool to use.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Is there a proof or reference off the top of your head that would demonstrate this being used?
    $endgroup$
    – Robert Wolfe
    Dec 4 '18 at 19:04






  • 1




    $begingroup$
    I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
    $endgroup$
    – Misha Lavrov
    Dec 4 '18 at 21:15





















6












$begingroup$

You can use Bernoulli's inequality in order to prove that $lim_{n rightarrow infty} sqrt[n]{a} =1$, where $a>0$. Here we define the $n$-th square root in elementary fashion by saying it is the solution of $x^n =a$. The existence can be shown by the Babylonian method or simply using statements on the existence of differentiable inverse functions.



Let $x_n +1 = sqrt[n]{a}$ for (w.l.o.g.) $a ge 1$. Then $(x_n+1)^n = a ge 1+nx_n$ and therefore $$frac{a}{n-1} ge x_n ge 0.$$
This proves $x_n rightarrow 0$. If $a< 1$, then we can apply the previous step with $b= 1/a$ and use that $sqrt{1/a} = 1/sqrt{a}$.



Another application: If we define the exponential function via $$exp(x) := lim_{n rightarrow infty} (1+x/n)^n,$$ then Bernoulli's inequality shows that $$exp(x) ge 1+x.$$






share|cite|improve this answer









$endgroup$





















    5












    $begingroup$

    I use it to prove that$$bigl(forall ain(0,infty)bigr):lim_{ntoinfty}sqrt[n]a=1.tag1$$It is clear that $(forall ninmathbb{N}):sqrt[n]a>1$. First, assume that $ageqslant1$. So, you can write $sqrt[n]a$ as $1+varepsilon_n(a)$, with $varepsilon_n(a)>0$ and $(1)$ is equivalent to$$lim_{ntoinfty}varepsilon_n(a)=0.tag2$$But now I can apply Bernoulli's inequality:begin{align}a&=left(sqrt[n]aright)^n\&=left(1+varepsilon_n(a)right)^n\&geqslant1+nvarepsilon_n(a)\&>nvarepsilon_n(a)end{align}and therefore $varepsilon_n(a)<frac an$. It follows then from the squeeze theorem that $(2)$ holds.



    Now, if $0<a<1$, then$$lim_{ntoinfty}sqrt[n]a=frac1{lim_{ntoinfty}sqrt[n]{frac1a}}=frac11=1.$$






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 16:41










    • $begingroup$
      Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
      $endgroup$
      – Pakk
      Dec 4 '18 at 21:04










    • $begingroup$
      @Pakk No. I wrote what I meant to write.
      $endgroup$
      – José Carlos Santos
      Dec 4 '18 at 21:06










    • $begingroup$
      If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
      $endgroup$
      – Pakk
      Dec 4 '18 at 21:10












    • $begingroup$
      I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
      $endgroup$
      – Pakk
      Dec 4 '18 at 21:19





















    2












    $begingroup$

    We can use Bernoulli's inequality to prove that the sequence
    $$
    a_n=Bigl(1+frac1nBigr)^n
    $$

    converges as $ntoinfty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
    begin{align*}
    frac{b_n}{b_{n-1}}
    &=frac{(1+frac1n)^{n+1}}{(1+frac1{n-1})^n}
    =frac{(n^2-1)^n(n+1)}{n^{2n}n}\
    &=frac{1+frac1n}{(1+frac1{n^2-1})^n}
    lefrac{1+frac1n}{1+frac n{n^2-1}}\
    &<frac{1+frac1n}{1+frac n{n^2}}
    =1.
    end{align*}

    Hence, $b_{n-1}>b_n$. Since $b_nge1$, $b_n$ converges as $ntoinfty$ which in turn implies that $a_n$ converges as $ntoinfty$ as well.






    share|cite|improve this answer









    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025737%2fwhere-does-one-even-need-bernoullis-inequality%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      5 Answers
      5






      active

      oldest

      votes








      5 Answers
      5






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      14












      $begingroup$

      Bernoulli's Inequality can prove the AM-GM Inequality. From this fact, you could derive Young's Inequality, Holder's Inequality, Minkowski's Inequality, and in turn any that follow from those.



      Let $a_1, a_2, ldots, a_n$ be $n$ positive real numbers. Let us define
      $$A_k=frac{a_1+a_2+cdots+a_k}{k}$$
      for every $1leq kleq n$. Bernoulli's Inequality in the form $x^kgeq 1+k(x-1)$ then implies
      $$left(frac{A_k}{A_{k-1}}right)^kgeq 1+kleft(frac{A_k}{A_{k-1}}-1right)$$
      which after some algebraic hyjinx results in
      $$A_k^kgeq a_kA_{k-1}^{k-1},.$$
      This in turn implies
      $$A_n^ngeq a_nA_{n-1}^{n-1}geq a_na_{n-1}A_{n-2}^{n-2}geq cdotsgeq a_ncdots a_2a_1$$
      which gives
      $$sqrt[n]{a_1cdots a_n}leqfrac{a_1+a_2+cdots+a_n}{n},.$$
      Intuitively, what's happening here is that we can order the values $A_1, A_2, ldots, A_n$ so that the subsequent quotients $A_k/A_{k-1}$ are close to $1$, which is where Bernoulli's is precise.






      share|cite|improve this answer









      $endgroup$









      • 2




        $begingroup$
        This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
        $endgroup$
        – Kezer
        Dec 4 '18 at 17:25






      • 4




        $begingroup$
        @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 17:47










      • $begingroup$
        Excellent intuition, thank you!
        $endgroup$
        – Kezer
        Dec 4 '18 at 19:14










      • $begingroup$
        Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 20:42






      • 1




        $begingroup$
        @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
        $endgroup$
        – Robert Wolfe
        Dec 5 '18 at 18:08
















      14












      $begingroup$

      Bernoulli's Inequality can prove the AM-GM Inequality. From this fact, you could derive Young's Inequality, Holder's Inequality, Minkowski's Inequality, and in turn any that follow from those.



      Let $a_1, a_2, ldots, a_n$ be $n$ positive real numbers. Let us define
      $$A_k=frac{a_1+a_2+cdots+a_k}{k}$$
      for every $1leq kleq n$. Bernoulli's Inequality in the form $x^kgeq 1+k(x-1)$ then implies
      $$left(frac{A_k}{A_{k-1}}right)^kgeq 1+kleft(frac{A_k}{A_{k-1}}-1right)$$
      which after some algebraic hyjinx results in
      $$A_k^kgeq a_kA_{k-1}^{k-1},.$$
      This in turn implies
      $$A_n^ngeq a_nA_{n-1}^{n-1}geq a_na_{n-1}A_{n-2}^{n-2}geq cdotsgeq a_ncdots a_2a_1$$
      which gives
      $$sqrt[n]{a_1cdots a_n}leqfrac{a_1+a_2+cdots+a_n}{n},.$$
      Intuitively, what's happening here is that we can order the values $A_1, A_2, ldots, A_n$ so that the subsequent quotients $A_k/A_{k-1}$ are close to $1$, which is where Bernoulli's is precise.






      share|cite|improve this answer









      $endgroup$









      • 2




        $begingroup$
        This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
        $endgroup$
        – Kezer
        Dec 4 '18 at 17:25






      • 4




        $begingroup$
        @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 17:47










      • $begingroup$
        Excellent intuition, thank you!
        $endgroup$
        – Kezer
        Dec 4 '18 at 19:14










      • $begingroup$
        Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 20:42






      • 1




        $begingroup$
        @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
        $endgroup$
        – Robert Wolfe
        Dec 5 '18 at 18:08














      14












      14








      14





      $begingroup$

      Bernoulli's Inequality can prove the AM-GM Inequality. From this fact, you could derive Young's Inequality, Holder's Inequality, Minkowski's Inequality, and in turn any that follow from those.



      Let $a_1, a_2, ldots, a_n$ be $n$ positive real numbers. Let us define
      $$A_k=frac{a_1+a_2+cdots+a_k}{k}$$
      for every $1leq kleq n$. Bernoulli's Inequality in the form $x^kgeq 1+k(x-1)$ then implies
      $$left(frac{A_k}{A_{k-1}}right)^kgeq 1+kleft(frac{A_k}{A_{k-1}}-1right)$$
      which after some algebraic hyjinx results in
      $$A_k^kgeq a_kA_{k-1}^{k-1},.$$
      This in turn implies
      $$A_n^ngeq a_nA_{n-1}^{n-1}geq a_na_{n-1}A_{n-2}^{n-2}geq cdotsgeq a_ncdots a_2a_1$$
      which gives
      $$sqrt[n]{a_1cdots a_n}leqfrac{a_1+a_2+cdots+a_n}{n},.$$
      Intuitively, what's happening here is that we can order the values $A_1, A_2, ldots, A_n$ so that the subsequent quotients $A_k/A_{k-1}$ are close to $1$, which is where Bernoulli's is precise.






      share|cite|improve this answer









      $endgroup$



      Bernoulli's Inequality can prove the AM-GM Inequality. From this fact, you could derive Young's Inequality, Holder's Inequality, Minkowski's Inequality, and in turn any that follow from those.



      Let $a_1, a_2, ldots, a_n$ be $n$ positive real numbers. Let us define
      $$A_k=frac{a_1+a_2+cdots+a_k}{k}$$
      for every $1leq kleq n$. Bernoulli's Inequality in the form $x^kgeq 1+k(x-1)$ then implies
      $$left(frac{A_k}{A_{k-1}}right)^kgeq 1+kleft(frac{A_k}{A_{k-1}}-1right)$$
      which after some algebraic hyjinx results in
      $$A_k^kgeq a_kA_{k-1}^{k-1},.$$
      This in turn implies
      $$A_n^ngeq a_nA_{n-1}^{n-1}geq a_na_{n-1}A_{n-2}^{n-2}geq cdotsgeq a_ncdots a_2a_1$$
      which gives
      $$sqrt[n]{a_1cdots a_n}leqfrac{a_1+a_2+cdots+a_n}{n},.$$
      Intuitively, what's happening here is that we can order the values $A_1, A_2, ldots, A_n$ so that the subsequent quotients $A_k/A_{k-1}$ are close to $1$, which is where Bernoulli's is precise.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Dec 4 '18 at 16:51









      Robert WolfeRobert Wolfe

      5,74422563




      5,74422563








      • 2




        $begingroup$
        This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
        $endgroup$
        – Kezer
        Dec 4 '18 at 17:25






      • 4




        $begingroup$
        @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 17:47










      • $begingroup$
        Excellent intuition, thank you!
        $endgroup$
        – Kezer
        Dec 4 '18 at 19:14










      • $begingroup$
        Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 20:42






      • 1




        $begingroup$
        @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
        $endgroup$
        – Robert Wolfe
        Dec 5 '18 at 18:08














      • 2




        $begingroup$
        This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
        $endgroup$
        – Kezer
        Dec 4 '18 at 17:25






      • 4




        $begingroup$
        @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 17:47










      • $begingroup$
        Excellent intuition, thank you!
        $endgroup$
        – Kezer
        Dec 4 '18 at 19:14










      • $begingroup$
        Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 20:42






      • 1




        $begingroup$
        @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
        $endgroup$
        – Robert Wolfe
        Dec 5 '18 at 18:08








      2




      2




      $begingroup$
      This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
      $endgroup$
      – Kezer
      Dec 4 '18 at 17:25




      $begingroup$
      This is a beautiful proof of AM-GM. I like your usage of "algebraic hyjinx" and that you included some intuition. I also like that the equality case of AM-GM comes out easily from this argument. I suppose yeah - that Bernoulli is, in fact, equivalent to AM-GM lets it serve as a strong tool!
      $endgroup$
      – Kezer
      Dec 4 '18 at 17:25




      4




      4




      $begingroup$
      @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 17:47




      $begingroup$
      @Kezer To take my earlier comment a little further, the reason why those three generate most inequalities is that their essential natures are pretty different. The triangle inequality could be thought of as the most important "ordering" inequality. Sums of squares are basically "guaranteed positives". Bernoulli's however is the first instance of "tangent inequality": i.e. the tangent line of $x^n$ at $1$ is $1+n(x-1)$. It's a nice coincidence that it can be proved without calculus.
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 17:47












      $begingroup$
      Excellent intuition, thank you!
      $endgroup$
      – Kezer
      Dec 4 '18 at 19:14




      $begingroup$
      Excellent intuition, thank you!
      $endgroup$
      – Kezer
      Dec 4 '18 at 19:14












      $begingroup$
      Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 20:42




      $begingroup$
      Hope this fixes your perception of Bernoulli's Inequality as "not a very strong inequality". It's an easy inequality. But powerful.
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 20:42




      1




      1




      $begingroup$
      @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
      $endgroup$
      – Robert Wolfe
      Dec 5 '18 at 18:08




      $begingroup$
      @Kezer You bring up a good point. A lot of inequalities because "obvious" after a certain point. For example, the AMGM inequality is "obvious" if the dispersion of the positive numbers is too high. In particular, if $n^n a_1leq a_n$ where $a_1$ an $a_n$ are the least and greatest element in the list. It is up to us to recast problems to use the optimal cases of inequalities.
      $endgroup$
      – Robert Wolfe
      Dec 5 '18 at 18:08











      8












      $begingroup$

      When dealing with asymptotics of probabilities, the expression $(1-p)^n$ comes up all the time. The most convenient way to handle it is with the inequalities $$1 - pn le (1-p)^n le e^{-pn}$$ where the lower bound is Bernoulli's inequality. (I'm assuming here that $p in [0,1]$ and $n$ is a natural number.) Actually, as mentioned in p4sch's answer, the upper bound is also a consequence of Bernoulli's inequality, via the inequality $1 + x le e^{x}$.



      For example, the result that monotone properties of the Erdős–Rényi model of random graphs have thresholds relies on the fact that if you take the union of $k$ copies of $mathcal G_{n,p}$, the graph you get (which has the distribution $mathcal G_{n,1-(1-p)^k}$) can be thought of as a subgraph of $mathcal G_{n,kp}$. This implies that as the edge probability $p$ scales linearly, the probability that your graph lacks a monotone property decays exponentially: $$Pr[mathcal G_{n,kp} text{ lacks property $M$}] le Pr[mathcal G_{n,p} text{ lacks property $M$}]^k.$$ For more details, see Theorem 1.7 in this textbook.



      Many inequalities can prove each other and it's hard to say you ever "need" a particular result. This, however, is an example where Bernoulli's inequality is the most convenient tool to use.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Is there a proof or reference off the top of your head that would demonstrate this being used?
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 19:04






      • 1




        $begingroup$
        I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
        $endgroup$
        – Misha Lavrov
        Dec 4 '18 at 21:15


















      8












      $begingroup$

      When dealing with asymptotics of probabilities, the expression $(1-p)^n$ comes up all the time. The most convenient way to handle it is with the inequalities $$1 - pn le (1-p)^n le e^{-pn}$$ where the lower bound is Bernoulli's inequality. (I'm assuming here that $p in [0,1]$ and $n$ is a natural number.) Actually, as mentioned in p4sch's answer, the upper bound is also a consequence of Bernoulli's inequality, via the inequality $1 + x le e^{x}$.



      For example, the result that monotone properties of the Erdős–Rényi model of random graphs have thresholds relies on the fact that if you take the union of $k$ copies of $mathcal G_{n,p}$, the graph you get (which has the distribution $mathcal G_{n,1-(1-p)^k}$) can be thought of as a subgraph of $mathcal G_{n,kp}$. This implies that as the edge probability $p$ scales linearly, the probability that your graph lacks a monotone property decays exponentially: $$Pr[mathcal G_{n,kp} text{ lacks property $M$}] le Pr[mathcal G_{n,p} text{ lacks property $M$}]^k.$$ For more details, see Theorem 1.7 in this textbook.



      Many inequalities can prove each other and it's hard to say you ever "need" a particular result. This, however, is an example where Bernoulli's inequality is the most convenient tool to use.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Is there a proof or reference off the top of your head that would demonstrate this being used?
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 19:04






      • 1




        $begingroup$
        I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
        $endgroup$
        – Misha Lavrov
        Dec 4 '18 at 21:15
















      8












      8








      8





      $begingroup$

      When dealing with asymptotics of probabilities, the expression $(1-p)^n$ comes up all the time. The most convenient way to handle it is with the inequalities $$1 - pn le (1-p)^n le e^{-pn}$$ where the lower bound is Bernoulli's inequality. (I'm assuming here that $p in [0,1]$ and $n$ is a natural number.) Actually, as mentioned in p4sch's answer, the upper bound is also a consequence of Bernoulli's inequality, via the inequality $1 + x le e^{x}$.



      For example, the result that monotone properties of the Erdős–Rényi model of random graphs have thresholds relies on the fact that if you take the union of $k$ copies of $mathcal G_{n,p}$, the graph you get (which has the distribution $mathcal G_{n,1-(1-p)^k}$) can be thought of as a subgraph of $mathcal G_{n,kp}$. This implies that as the edge probability $p$ scales linearly, the probability that your graph lacks a monotone property decays exponentially: $$Pr[mathcal G_{n,kp} text{ lacks property $M$}] le Pr[mathcal G_{n,p} text{ lacks property $M$}]^k.$$ For more details, see Theorem 1.7 in this textbook.



      Many inequalities can prove each other and it's hard to say you ever "need" a particular result. This, however, is an example where Bernoulli's inequality is the most convenient tool to use.






      share|cite|improve this answer











      $endgroup$



      When dealing with asymptotics of probabilities, the expression $(1-p)^n$ comes up all the time. The most convenient way to handle it is with the inequalities $$1 - pn le (1-p)^n le e^{-pn}$$ where the lower bound is Bernoulli's inequality. (I'm assuming here that $p in [0,1]$ and $n$ is a natural number.) Actually, as mentioned in p4sch's answer, the upper bound is also a consequence of Bernoulli's inequality, via the inequality $1 + x le e^{x}$.



      For example, the result that monotone properties of the Erdős–Rényi model of random graphs have thresholds relies on the fact that if you take the union of $k$ copies of $mathcal G_{n,p}$, the graph you get (which has the distribution $mathcal G_{n,1-(1-p)^k}$) can be thought of as a subgraph of $mathcal G_{n,kp}$. This implies that as the edge probability $p$ scales linearly, the probability that your graph lacks a monotone property decays exponentially: $$Pr[mathcal G_{n,kp} text{ lacks property $M$}] le Pr[mathcal G_{n,p} text{ lacks property $M$}]^k.$$ For more details, see Theorem 1.7 in this textbook.



      Many inequalities can prove each other and it's hard to say you ever "need" a particular result. This, however, is an example where Bernoulli's inequality is the most convenient tool to use.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 4 '18 at 21:26

























      answered Dec 4 '18 at 18:51









      Misha LavrovMisha Lavrov

      44.9k556107




      44.9k556107












      • $begingroup$
        Is there a proof or reference off the top of your head that would demonstrate this being used?
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 19:04






      • 1




        $begingroup$
        I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
        $endgroup$
        – Misha Lavrov
        Dec 4 '18 at 21:15




















      • $begingroup$
        Is there a proof or reference off the top of your head that would demonstrate this being used?
        $endgroup$
        – Robert Wolfe
        Dec 4 '18 at 19:04






      • 1




        $begingroup$
        I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
        $endgroup$
        – Misha Lavrov
        Dec 4 '18 at 21:15


















      $begingroup$
      Is there a proof or reference off the top of your head that would demonstrate this being used?
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 19:04




      $begingroup$
      Is there a proof or reference off the top of your head that would demonstrate this being used?
      $endgroup$
      – Robert Wolfe
      Dec 4 '18 at 19:04




      1




      1




      $begingroup$
      I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
      $endgroup$
      – Misha Lavrov
      Dec 4 '18 at 21:15






      $begingroup$
      I will edit the answer with an example theorem about random graphs, which is Theorem 1.7 in Introduction to Random Graphs by Frieze and Karonski. But this is just the first example I found from that textbook, and there are plenty more. (Bernoulli's inequality is not called out by name when the authors of this textbook use it.)
      $endgroup$
      – Misha Lavrov
      Dec 4 '18 at 21:15













      6












      $begingroup$

      You can use Bernoulli's inequality in order to prove that $lim_{n rightarrow infty} sqrt[n]{a} =1$, where $a>0$. Here we define the $n$-th square root in elementary fashion by saying it is the solution of $x^n =a$. The existence can be shown by the Babylonian method or simply using statements on the existence of differentiable inverse functions.



      Let $x_n +1 = sqrt[n]{a}$ for (w.l.o.g.) $a ge 1$. Then $(x_n+1)^n = a ge 1+nx_n$ and therefore $$frac{a}{n-1} ge x_n ge 0.$$
      This proves $x_n rightarrow 0$. If $a< 1$, then we can apply the previous step with $b= 1/a$ and use that $sqrt{1/a} = 1/sqrt{a}$.



      Another application: If we define the exponential function via $$exp(x) := lim_{n rightarrow infty} (1+x/n)^n,$$ then Bernoulli's inequality shows that $$exp(x) ge 1+x.$$






      share|cite|improve this answer









      $endgroup$


















        6












        $begingroup$

        You can use Bernoulli's inequality in order to prove that $lim_{n rightarrow infty} sqrt[n]{a} =1$, where $a>0$. Here we define the $n$-th square root in elementary fashion by saying it is the solution of $x^n =a$. The existence can be shown by the Babylonian method or simply using statements on the existence of differentiable inverse functions.



        Let $x_n +1 = sqrt[n]{a}$ for (w.l.o.g.) $a ge 1$. Then $(x_n+1)^n = a ge 1+nx_n$ and therefore $$frac{a}{n-1} ge x_n ge 0.$$
        This proves $x_n rightarrow 0$. If $a< 1$, then we can apply the previous step with $b= 1/a$ and use that $sqrt{1/a} = 1/sqrt{a}$.



        Another application: If we define the exponential function via $$exp(x) := lim_{n rightarrow infty} (1+x/n)^n,$$ then Bernoulli's inequality shows that $$exp(x) ge 1+x.$$






        share|cite|improve this answer









        $endgroup$
















          6












          6








          6





          $begingroup$

          You can use Bernoulli's inequality in order to prove that $lim_{n rightarrow infty} sqrt[n]{a} =1$, where $a>0$. Here we define the $n$-th square root in elementary fashion by saying it is the solution of $x^n =a$. The existence can be shown by the Babylonian method or simply using statements on the existence of differentiable inverse functions.



          Let $x_n +1 = sqrt[n]{a}$ for (w.l.o.g.) $a ge 1$. Then $(x_n+1)^n = a ge 1+nx_n$ and therefore $$frac{a}{n-1} ge x_n ge 0.$$
          This proves $x_n rightarrow 0$. If $a< 1$, then we can apply the previous step with $b= 1/a$ and use that $sqrt{1/a} = 1/sqrt{a}$.



          Another application: If we define the exponential function via $$exp(x) := lim_{n rightarrow infty} (1+x/n)^n,$$ then Bernoulli's inequality shows that $$exp(x) ge 1+x.$$






          share|cite|improve this answer









          $endgroup$



          You can use Bernoulli's inequality in order to prove that $lim_{n rightarrow infty} sqrt[n]{a} =1$, where $a>0$. Here we define the $n$-th square root in elementary fashion by saying it is the solution of $x^n =a$. The existence can be shown by the Babylonian method or simply using statements on the existence of differentiable inverse functions.



          Let $x_n +1 = sqrt[n]{a}$ for (w.l.o.g.) $a ge 1$. Then $(x_n+1)^n = a ge 1+nx_n$ and therefore $$frac{a}{n-1} ge x_n ge 0.$$
          This proves $x_n rightarrow 0$. If $a< 1$, then we can apply the previous step with $b= 1/a$ and use that $sqrt{1/a} = 1/sqrt{a}$.



          Another application: If we define the exponential function via $$exp(x) := lim_{n rightarrow infty} (1+x/n)^n,$$ then Bernoulli's inequality shows that $$exp(x) ge 1+x.$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 16:11









          p4schp4sch

          4,995217




          4,995217























              5












              $begingroup$

              I use it to prove that$$bigl(forall ain(0,infty)bigr):lim_{ntoinfty}sqrt[n]a=1.tag1$$It is clear that $(forall ninmathbb{N}):sqrt[n]a>1$. First, assume that $ageqslant1$. So, you can write $sqrt[n]a$ as $1+varepsilon_n(a)$, with $varepsilon_n(a)>0$ and $(1)$ is equivalent to$$lim_{ntoinfty}varepsilon_n(a)=0.tag2$$But now I can apply Bernoulli's inequality:begin{align}a&=left(sqrt[n]aright)^n\&=left(1+varepsilon_n(a)right)^n\&geqslant1+nvarepsilon_n(a)\&>nvarepsilon_n(a)end{align}and therefore $varepsilon_n(a)<frac an$. It follows then from the squeeze theorem that $(2)$ holds.



              Now, if $0<a<1$, then$$lim_{ntoinfty}sqrt[n]a=frac1{lim_{ntoinfty}sqrt[n]{frac1a}}=frac11=1.$$






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
                $endgroup$
                – Robert Wolfe
                Dec 4 '18 at 16:41










              • $begingroup$
                Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:04










              • $begingroup$
                @Pakk No. I wrote what I meant to write.
                $endgroup$
                – José Carlos Santos
                Dec 4 '18 at 21:06










              • $begingroup$
                If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:10












              • $begingroup$
                I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:19


















              5












              $begingroup$

              I use it to prove that$$bigl(forall ain(0,infty)bigr):lim_{ntoinfty}sqrt[n]a=1.tag1$$It is clear that $(forall ninmathbb{N}):sqrt[n]a>1$. First, assume that $ageqslant1$. So, you can write $sqrt[n]a$ as $1+varepsilon_n(a)$, with $varepsilon_n(a)>0$ and $(1)$ is equivalent to$$lim_{ntoinfty}varepsilon_n(a)=0.tag2$$But now I can apply Bernoulli's inequality:begin{align}a&=left(sqrt[n]aright)^n\&=left(1+varepsilon_n(a)right)^n\&geqslant1+nvarepsilon_n(a)\&>nvarepsilon_n(a)end{align}and therefore $varepsilon_n(a)<frac an$. It follows then from the squeeze theorem that $(2)$ holds.



              Now, if $0<a<1$, then$$lim_{ntoinfty}sqrt[n]a=frac1{lim_{ntoinfty}sqrt[n]{frac1a}}=frac11=1.$$






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
                $endgroup$
                – Robert Wolfe
                Dec 4 '18 at 16:41










              • $begingroup$
                Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:04










              • $begingroup$
                @Pakk No. I wrote what I meant to write.
                $endgroup$
                – José Carlos Santos
                Dec 4 '18 at 21:06










              • $begingroup$
                If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:10












              • $begingroup$
                I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:19
















              5












              5








              5





              $begingroup$

              I use it to prove that$$bigl(forall ain(0,infty)bigr):lim_{ntoinfty}sqrt[n]a=1.tag1$$It is clear that $(forall ninmathbb{N}):sqrt[n]a>1$. First, assume that $ageqslant1$. So, you can write $sqrt[n]a$ as $1+varepsilon_n(a)$, with $varepsilon_n(a)>0$ and $(1)$ is equivalent to$$lim_{ntoinfty}varepsilon_n(a)=0.tag2$$But now I can apply Bernoulli's inequality:begin{align}a&=left(sqrt[n]aright)^n\&=left(1+varepsilon_n(a)right)^n\&geqslant1+nvarepsilon_n(a)\&>nvarepsilon_n(a)end{align}and therefore $varepsilon_n(a)<frac an$. It follows then from the squeeze theorem that $(2)$ holds.



              Now, if $0<a<1$, then$$lim_{ntoinfty}sqrt[n]a=frac1{lim_{ntoinfty}sqrt[n]{frac1a}}=frac11=1.$$






              share|cite|improve this answer











              $endgroup$



              I use it to prove that$$bigl(forall ain(0,infty)bigr):lim_{ntoinfty}sqrt[n]a=1.tag1$$It is clear that $(forall ninmathbb{N}):sqrt[n]a>1$. First, assume that $ageqslant1$. So, you can write $sqrt[n]a$ as $1+varepsilon_n(a)$, with $varepsilon_n(a)>0$ and $(1)$ is equivalent to$$lim_{ntoinfty}varepsilon_n(a)=0.tag2$$But now I can apply Bernoulli's inequality:begin{align}a&=left(sqrt[n]aright)^n\&=left(1+varepsilon_n(a)right)^n\&geqslant1+nvarepsilon_n(a)\&>nvarepsilon_n(a)end{align}and therefore $varepsilon_n(a)<frac an$. It follows then from the squeeze theorem that $(2)$ holds.



              Now, if $0<a<1$, then$$lim_{ntoinfty}sqrt[n]a=frac1{lim_{ntoinfty}sqrt[n]{frac1a}}=frac11=1.$$







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Dec 4 '18 at 21:31

























              answered Dec 4 '18 at 16:11









              José Carlos SantosJosé Carlos Santos

              155k22124227




              155k22124227








              • 1




                $begingroup$
                I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
                $endgroup$
                – Robert Wolfe
                Dec 4 '18 at 16:41










              • $begingroup$
                Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:04










              • $begingroup$
                @Pakk No. I wrote what I meant to write.
                $endgroup$
                – José Carlos Santos
                Dec 4 '18 at 21:06










              • $begingroup$
                If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:10












              • $begingroup$
                I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:19
















              • 1




                $begingroup$
                I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
                $endgroup$
                – Robert Wolfe
                Dec 4 '18 at 16:41










              • $begingroup$
                Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:04










              • $begingroup$
                @Pakk No. I wrote what I meant to write.
                $endgroup$
                – José Carlos Santos
                Dec 4 '18 at 21:06










              • $begingroup$
                If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:10












              • $begingroup$
                I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
                $endgroup$
                – Pakk
                Dec 4 '18 at 21:19










              1




              1




              $begingroup$
              I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
              $endgroup$
              – Robert Wolfe
              Dec 4 '18 at 16:41




              $begingroup$
              I'm rather found of the quite explicit bounds $$frac{1-a^{-1}}{n}leq sqrt[n]{a}-1leqfrac{a-1}{n},.$$
              $endgroup$
              – Robert Wolfe
              Dec 4 '18 at 16:41












              $begingroup$
              Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:04




              $begingroup$
              Where you wrote $sqrt[n]{a}>1$ and $epsilon_n(a)>0$, I think you mean $sqrt[n]{a}>0$ and $epsilon_n(a)>-1$.
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:04












              $begingroup$
              @Pakk No. I wrote what I meant to write.
              $endgroup$
              – José Carlos Santos
              Dec 4 '18 at 21:06




              $begingroup$
              @Pakk No. I wrote what I meant to write.
              $endgroup$
              – José Carlos Santos
              Dec 4 '18 at 21:06












              $begingroup$
              If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:10






              $begingroup$
              If I take $a=0.25$ and $n=2$, then $sqrt{0.25}=0.5<1$, not larger than one...
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:10














              $begingroup$
              I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:19






              $begingroup$
              I think that the current version of your proof only works for $a ge 1$, but it is easily repaired to work for $a > 0$. Apologies if I miss something obvious.
              $endgroup$
              – Pakk
              Dec 4 '18 at 21:19













              2












              $begingroup$

              We can use Bernoulli's inequality to prove that the sequence
              $$
              a_n=Bigl(1+frac1nBigr)^n
              $$

              converges as $ntoinfty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
              begin{align*}
              frac{b_n}{b_{n-1}}
              &=frac{(1+frac1n)^{n+1}}{(1+frac1{n-1})^n}
              =frac{(n^2-1)^n(n+1)}{n^{2n}n}\
              &=frac{1+frac1n}{(1+frac1{n^2-1})^n}
              lefrac{1+frac1n}{1+frac n{n^2-1}}\
              &<frac{1+frac1n}{1+frac n{n^2}}
              =1.
              end{align*}

              Hence, $b_{n-1}>b_n$. Since $b_nge1$, $b_n$ converges as $ntoinfty$ which in turn implies that $a_n$ converges as $ntoinfty$ as well.






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                We can use Bernoulli's inequality to prove that the sequence
                $$
                a_n=Bigl(1+frac1nBigr)^n
                $$

                converges as $ntoinfty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
                begin{align*}
                frac{b_n}{b_{n-1}}
                &=frac{(1+frac1n)^{n+1}}{(1+frac1{n-1})^n}
                =frac{(n^2-1)^n(n+1)}{n^{2n}n}\
                &=frac{1+frac1n}{(1+frac1{n^2-1})^n}
                lefrac{1+frac1n}{1+frac n{n^2-1}}\
                &<frac{1+frac1n}{1+frac n{n^2}}
                =1.
                end{align*}

                Hence, $b_{n-1}>b_n$. Since $b_nge1$, $b_n$ converges as $ntoinfty$ which in turn implies that $a_n$ converges as $ntoinfty$ as well.






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  We can use Bernoulli's inequality to prove that the sequence
                  $$
                  a_n=Bigl(1+frac1nBigr)^n
                  $$

                  converges as $ntoinfty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
                  begin{align*}
                  frac{b_n}{b_{n-1}}
                  &=frac{(1+frac1n)^{n+1}}{(1+frac1{n-1})^n}
                  =frac{(n^2-1)^n(n+1)}{n^{2n}n}\
                  &=frac{1+frac1n}{(1+frac1{n^2-1})^n}
                  lefrac{1+frac1n}{1+frac n{n^2-1}}\
                  &<frac{1+frac1n}{1+frac n{n^2}}
                  =1.
                  end{align*}

                  Hence, $b_{n-1}>b_n$. Since $b_nge1$, $b_n$ converges as $ntoinfty$ which in turn implies that $a_n$ converges as $ntoinfty$ as well.






                  share|cite|improve this answer









                  $endgroup$



                  We can use Bernoulli's inequality to prove that the sequence
                  $$
                  a_n=Bigl(1+frac1nBigr)^n
                  $$

                  converges as $ntoinfty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
                  begin{align*}
                  frac{b_n}{b_{n-1}}
                  &=frac{(1+frac1n)^{n+1}}{(1+frac1{n-1})^n}
                  =frac{(n^2-1)^n(n+1)}{n^{2n}n}\
                  &=frac{1+frac1n}{(1+frac1{n^2-1})^n}
                  lefrac{1+frac1n}{1+frac n{n^2-1}}\
                  &<frac{1+frac1n}{1+frac n{n^2}}
                  =1.
                  end{align*}

                  Hence, $b_{n-1}>b_n$. Since $b_nge1$, $b_n$ converges as $ntoinfty$ which in turn implies that $a_n$ converges as $ntoinfty$ as well.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 5 '18 at 13:47









                  Cm7F7BbCm7F7Bb

                  12.4k32242




                  12.4k32242






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025737%2fwhere-does-one-even-need-bernoullis-inequality%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      Probability when a professor distributes a quiz and homework assignment to a class of n students.

                      Aardman Animations

                      Are they similar matrix