How to use the terms “fixed”, “arbitrary”, “fixed but arbitrary”, “given”, “for all” …...












5












$begingroup$


When I initially moved from Applied Mathematics background to Pure Mathematics (Graduate school), things were very tough in terms of proofs but they have gotten much better now. At those times, I couldn't explain what the following meant in proofs;




  1. Let $epsilon>0$ be given;

  2. Let $ninBbb{N}$ be fixed;

  3. Let $xinBbb{R}$ be fixed but arbitrary;

  4. Take $n=N+1,$ for some $NinBbb{N}$;

  5. In particular, for $epsilon=1/2$;

  6. Let $epsilon'>0$, for $epsilon=epsilon'/3$

  7. Such that;

  8. Whenever,

  9. For each;

  10. For all, and many more.


Could you please, explain these in details, perhaps, with some examples? That is, when to use them and how to... If you have other examples, I'd be glad to learn from you. Thanks










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Related: When do I use “arbitrary” and/or “fixed” in a proof?.
    $endgroup$
    – StackTD
    Jan 4 at 15:00






  • 2




    $begingroup$
    These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
    $endgroup$
    – littleO
    Jan 4 at 15:09






  • 6




    $begingroup$
    These fixed terms are somewhat arbitrary ...
    $endgroup$
    – Hagen von Eitzen
    Jan 4 at 15:40






  • 2




    $begingroup$
    "In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
    $endgroup$
    – timtfj
    Jan 4 at 15:44








  • 2




    $begingroup$
    "Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
    $endgroup$
    – timtfj
    Jan 4 at 15:48
















5












$begingroup$


When I initially moved from Applied Mathematics background to Pure Mathematics (Graduate school), things were very tough in terms of proofs but they have gotten much better now. At those times, I couldn't explain what the following meant in proofs;




  1. Let $epsilon>0$ be given;

  2. Let $ninBbb{N}$ be fixed;

  3. Let $xinBbb{R}$ be fixed but arbitrary;

  4. Take $n=N+1,$ for some $NinBbb{N}$;

  5. In particular, for $epsilon=1/2$;

  6. Let $epsilon'>0$, for $epsilon=epsilon'/3$

  7. Such that;

  8. Whenever,

  9. For each;

  10. For all, and many more.


Could you please, explain these in details, perhaps, with some examples? That is, when to use them and how to... If you have other examples, I'd be glad to learn from you. Thanks










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Related: When do I use “arbitrary” and/or “fixed” in a proof?.
    $endgroup$
    – StackTD
    Jan 4 at 15:00






  • 2




    $begingroup$
    These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
    $endgroup$
    – littleO
    Jan 4 at 15:09






  • 6




    $begingroup$
    These fixed terms are somewhat arbitrary ...
    $endgroup$
    – Hagen von Eitzen
    Jan 4 at 15:40






  • 2




    $begingroup$
    "In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
    $endgroup$
    – timtfj
    Jan 4 at 15:44








  • 2




    $begingroup$
    "Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
    $endgroup$
    – timtfj
    Jan 4 at 15:48














5












5








5


1



$begingroup$


When I initially moved from Applied Mathematics background to Pure Mathematics (Graduate school), things were very tough in terms of proofs but they have gotten much better now. At those times, I couldn't explain what the following meant in proofs;




  1. Let $epsilon>0$ be given;

  2. Let $ninBbb{N}$ be fixed;

  3. Let $xinBbb{R}$ be fixed but arbitrary;

  4. Take $n=N+1,$ for some $NinBbb{N}$;

  5. In particular, for $epsilon=1/2$;

  6. Let $epsilon'>0$, for $epsilon=epsilon'/3$

  7. Such that;

  8. Whenever,

  9. For each;

  10. For all, and many more.


Could you please, explain these in details, perhaps, with some examples? That is, when to use them and how to... If you have other examples, I'd be glad to learn from you. Thanks










share|cite|improve this question











$endgroup$




When I initially moved from Applied Mathematics background to Pure Mathematics (Graduate school), things were very tough in terms of proofs but they have gotten much better now. At those times, I couldn't explain what the following meant in proofs;




  1. Let $epsilon>0$ be given;

  2. Let $ninBbb{N}$ be fixed;

  3. Let $xinBbb{R}$ be fixed but arbitrary;

  4. Take $n=N+1,$ for some $NinBbb{N}$;

  5. In particular, for $epsilon=1/2$;

  6. Let $epsilon'>0$, for $epsilon=epsilon'/3$

  7. Such that;

  8. Whenever,

  9. For each;

  10. For all, and many more.


Could you please, explain these in details, perhaps, with some examples? That is, when to use them and how to... If you have other examples, I'd be glad to learn from you. Thanks







real-analysis proof-verification proof-writing terminology proof-explanation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 6:19







Omojola Micheal

















asked Jan 4 at 14:58









Omojola MichealOmojola Micheal

2,013424




2,013424








  • 2




    $begingroup$
    Related: When do I use “arbitrary” and/or “fixed” in a proof?.
    $endgroup$
    – StackTD
    Jan 4 at 15:00






  • 2




    $begingroup$
    These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
    $endgroup$
    – littleO
    Jan 4 at 15:09






  • 6




    $begingroup$
    These fixed terms are somewhat arbitrary ...
    $endgroup$
    – Hagen von Eitzen
    Jan 4 at 15:40






  • 2




    $begingroup$
    "In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
    $endgroup$
    – timtfj
    Jan 4 at 15:44








  • 2




    $begingroup$
    "Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
    $endgroup$
    – timtfj
    Jan 4 at 15:48














  • 2




    $begingroup$
    Related: When do I use “arbitrary” and/or “fixed” in a proof?.
    $endgroup$
    – StackTD
    Jan 4 at 15:00






  • 2




    $begingroup$
    These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
    $endgroup$
    – littleO
    Jan 4 at 15:09






  • 6




    $begingroup$
    These fixed terms are somewhat arbitrary ...
    $endgroup$
    – Hagen von Eitzen
    Jan 4 at 15:40






  • 2




    $begingroup$
    "In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
    $endgroup$
    – timtfj
    Jan 4 at 15:44








  • 2




    $begingroup$
    "Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
    $endgroup$
    – timtfj
    Jan 4 at 15:48








2




2




$begingroup$
Related: When do I use “arbitrary” and/or “fixed” in a proof?.
$endgroup$
– StackTD
Jan 4 at 15:00




$begingroup$
Related: When do I use “arbitrary” and/or “fixed” in a proof?.
$endgroup$
– StackTD
Jan 4 at 15:00




2




2




$begingroup$
These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
$endgroup$
– littleO
Jan 4 at 15:09




$begingroup$
These four statements could be replaced with: 1. Let $epsilon > 0$. 2. Let $n in mathbb N$. 3. Let $x in mathbb R$. 4. Let $n = N + 1$.
$endgroup$
– littleO
Jan 4 at 15:09




6




6




$begingroup$
These fixed terms are somewhat arbitrary ...
$endgroup$
– Hagen von Eitzen
Jan 4 at 15:40




$begingroup$
These fixed terms are somewhat arbitrary ...
$endgroup$
– Hagen von Eitzen
Jan 4 at 15:40




2




2




$begingroup$
"In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
$endgroup$
– timtfj
Jan 4 at 15:44






$begingroup$
"In particular, for $x=3$: we're going to temporarily let $x=3$ to examine that case, but $x$ is still a variable and not defined to be always $3$.
$endgroup$
– timtfj
Jan 4 at 15:44






2




2




$begingroup$
"Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
$endgroup$
– timtfj
Jan 4 at 15:48




$begingroup$
"Fixed but arbitrary" we can choose any value we like, but once chisen, it's a constant.
$endgroup$
– timtfj
Jan 4 at 15:48










1 Answer
1






active

oldest

votes


















2












$begingroup$

1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$



PROOF



Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}



2. Let $ninBbb{N}$ be fixed;



Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.



EXAMPLE 2



Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



PROOF



Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



3. Such that, arbitrary



EXAMPLE 3



Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if



begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF



Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.



4. Fixed but arbitrary, fixed, such that and whenever.



EXAMPLE 4



Prove that $|x|$ is continuous on $Bbb{R}$.



Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
    $endgroup$
    – Todor Markov
    Jan 4 at 16:37










  • $begingroup$
    @Todor Markov: They, can be but I was trying to find other words of expressing them
    $endgroup$
    – Omojola Micheal
    Jan 4 at 19:15












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1 Answer
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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$



PROOF



Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}



2. Let $ninBbb{N}$ be fixed;



Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.



EXAMPLE 2



Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



PROOF



Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



3. Such that, arbitrary



EXAMPLE 3



Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if



begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF



Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.



4. Fixed but arbitrary, fixed, such that and whenever.



EXAMPLE 4



Prove that $|x|$ is continuous on $Bbb{R}$.



Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
    $endgroup$
    – Todor Markov
    Jan 4 at 16:37










  • $begingroup$
    @Todor Markov: They, can be but I was trying to find other words of expressing them
    $endgroup$
    – Omojola Micheal
    Jan 4 at 19:15
















2












$begingroup$

1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$



PROOF



Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}



2. Let $ninBbb{N}$ be fixed;



Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.



EXAMPLE 2



Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



PROOF



Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



3. Such that, arbitrary



EXAMPLE 3



Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if



begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF



Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.



4. Fixed but arbitrary, fixed, such that and whenever.



EXAMPLE 4



Prove that $|x|$ is continuous on $Bbb{R}$.



Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
    $endgroup$
    – Todor Markov
    Jan 4 at 16:37










  • $begingroup$
    @Todor Markov: They, can be but I was trying to find other words of expressing them
    $endgroup$
    – Omojola Micheal
    Jan 4 at 19:15














2












2








2





$begingroup$

1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$



PROOF



Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}



2. Let $ninBbb{N}$ be fixed;



Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.



EXAMPLE 2



Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



PROOF



Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



3. Such that, arbitrary



EXAMPLE 3



Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if



begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF



Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.



4. Fixed but arbitrary, fixed, such that and whenever.



EXAMPLE 4



Prove that $|x|$ is continuous on $Bbb{R}$.



Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.






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$endgroup$



1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$



PROOF



Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}



2. Let $ninBbb{N}$ be fixed;



Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.



EXAMPLE 2



Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



PROOF



Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}



3. Such that, arbitrary



EXAMPLE 3



Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if



begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF



Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.



4. Fixed but arbitrary, fixed, such that and whenever.



EXAMPLE 4



Prove that $|x|$ is continuous on $Bbb{R}$.



Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 4 at 19:35

























answered Jan 4 at 16:01









Omojola MichealOmojola Micheal

2,013424




2,013424












  • $begingroup$
    Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
    $endgroup$
    – Todor Markov
    Jan 4 at 16:37










  • $begingroup$
    @Todor Markov: They, can be but I was trying to find other words of expressing them
    $endgroup$
    – Omojola Micheal
    Jan 4 at 19:15


















  • $begingroup$
    Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
    $endgroup$
    – Todor Markov
    Jan 4 at 16:37










  • $begingroup$
    @Todor Markov: They, can be but I was trying to find other words of expressing them
    $endgroup$
    – Omojola Micheal
    Jan 4 at 19:15
















$begingroup$
Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
$endgroup$
– Todor Markov
Jan 4 at 16:37




$begingroup$
Do you know of cases where given (from point 1), fixed (from point 2), and fixed but arbitrary (from point 3) are NOT interchangeable?
$endgroup$
– Todor Markov
Jan 4 at 16:37












$begingroup$
@Todor Markov: They, can be but I was trying to find other words of expressing them
$endgroup$
– Omojola Micheal
Jan 4 at 19:15




$begingroup$
@Todor Markov: They, can be but I was trying to find other words of expressing them
$endgroup$
– Omojola Micheal
Jan 4 at 19:15


















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