Show that if $inf(A^+)=a>0$ then $ain A$ and $A={za;zin mathbb{Z}}$
$begingroup$
$A$ is a set such that $x,yin ARightarrow$ $x-yin A$, and $A^+$ is a subset of $A$ which contains only its positive elements.
I was able to successfully show that if $nain A$, $ninmathbb{Z}, n neq 0$ then $(na,na pm a)cap A=emptyset$ and that if $infA^+=0$ then $A$ is dense in $mathbb{R}$.
Not sure if the first the latter is useful in this context but I am leaving it here in case it is (maybe consider set $A-a$?).
Now, the only thing left for me to prove is that all $zain A$ for $zin mathbb{Z}$. From what I've observed, it suffices to show that $ain A$, but I have been stuck trying to prove that for a while with no success whatsoever.
If anyone could offer some help it would be highly appreciated. Please do not just give out the answer though. Some tip/insight to get me on the right track would be more interesting.
real-analysis
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
$endgroup$
add a comment |
$begingroup$
$A$ is a set such that $x,yin ARightarrow$ $x-yin A$, and $A^+$ is a subset of $A$ which contains only its positive elements.
I was able to successfully show that if $nain A$, $ninmathbb{Z}, n neq 0$ then $(na,na pm a)cap A=emptyset$ and that if $infA^+=0$ then $A$ is dense in $mathbb{R}$.
Not sure if the first the latter is useful in this context but I am leaving it here in case it is (maybe consider set $A-a$?).
Now, the only thing left for me to prove is that all $zain A$ for $zin mathbb{Z}$. From what I've observed, it suffices to show that $ain A$, but I have been stuck trying to prove that for a while with no success whatsoever.
If anyone could offer some help it would be highly appreciated. Please do not just give out the answer though. Some tip/insight to get me on the right track would be more interesting.
real-analysis
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
$endgroup$
add a comment |
$begingroup$
$A$ is a set such that $x,yin ARightarrow$ $x-yin A$, and $A^+$ is a subset of $A$ which contains only its positive elements.
I was able to successfully show that if $nain A$, $ninmathbb{Z}, n neq 0$ then $(na,na pm a)cap A=emptyset$ and that if $infA^+=0$ then $A$ is dense in $mathbb{R}$.
Not sure if the first the latter is useful in this context but I am leaving it here in case it is (maybe consider set $A-a$?).
Now, the only thing left for me to prove is that all $zain A$ for $zin mathbb{Z}$. From what I've observed, it suffices to show that $ain A$, but I have been stuck trying to prove that for a while with no success whatsoever.
If anyone could offer some help it would be highly appreciated. Please do not just give out the answer though. Some tip/insight to get me on the right track would be more interesting.
real-analysis
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
$endgroup$
$A$ is a set such that $x,yin ARightarrow$ $x-yin A$, and $A^+$ is a subset of $A$ which contains only its positive elements.
I was able to successfully show that if $nain A$, $ninmathbb{Z}, n neq 0$ then $(na,na pm a)cap A=emptyset$ and that if $infA^+=0$ then $A$ is dense in $mathbb{R}$.
Not sure if the first the latter is useful in this context but I am leaving it here in case it is (maybe consider set $A-a$?).
Now, the only thing left for me to prove is that all $zain A$ for $zin mathbb{Z}$. From what I've observed, it suffices to show that $ain A$, but I have been stuck trying to prove that for a while with no success whatsoever.
If anyone could offer some help it would be highly appreciated. Please do not just give out the answer though. Some tip/insight to get me on the right track would be more interesting.
real-analysis
real-analysis
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
asked Dec 10 '18 at 15:11
PolygonsPolygons
82
82
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Observe: $A$ is a group with addition: Assume it is nonempty so $exists xin A$.
i) $x-x=0 in A$
ii) If $xin A$ then $0-x in A$
iii) If $x,yin A$ then $x+y=x-(-y)in A$
Now the statement is basically saying discrete subgroups of $mathbb{R}$ is cyclic:
Suppose your chosen $anotin A$. Then there must be a sequence in $A$: $a_ndownarrow a$ by definition of $inf$. Now consider the sequence $(a_n-a_m)_{m,n}$ to get a contradiction. (A convergent sequence is Cauchy)
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
$endgroup$
add a comment |
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$begingroup$
Observe: $A$ is a group with addition: Assume it is nonempty so $exists xin A$.
i) $x-x=0 in A$
ii) If $xin A$ then $0-x in A$
iii) If $x,yin A$ then $x+y=x-(-y)in A$
Now the statement is basically saying discrete subgroups of $mathbb{R}$ is cyclic:
Suppose your chosen $anotin A$. Then there must be a sequence in $A$: $a_ndownarrow a$ by definition of $inf$. Now consider the sequence $(a_n-a_m)_{m,n}$ to get a contradiction. (A convergent sequence is Cauchy)
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
$endgroup$
add a comment |
$begingroup$
Observe: $A$ is a group with addition: Assume it is nonempty so $exists xin A$.
i) $x-x=0 in A$
ii) If $xin A$ then $0-x in A$
iii) If $x,yin A$ then $x+y=x-(-y)in A$
Now the statement is basically saying discrete subgroups of $mathbb{R}$ is cyclic:
Suppose your chosen $anotin A$. Then there must be a sequence in $A$: $a_ndownarrow a$ by definition of $inf$. Now consider the sequence $(a_n-a_m)_{m,n}$ to get a contradiction. (A convergent sequence is Cauchy)
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
$endgroup$
add a comment |
$begingroup$
Observe: $A$ is a group with addition: Assume it is nonempty so $exists xin A$.
i) $x-x=0 in A$
ii) If $xin A$ then $0-x in A$
iii) If $x,yin A$ then $x+y=x-(-y)in A$
Now the statement is basically saying discrete subgroups of $mathbb{R}$ is cyclic:
Suppose your chosen $anotin A$. Then there must be a sequence in $A$: $a_ndownarrow a$ by definition of $inf$. Now consider the sequence $(a_n-a_m)_{m,n}$ to get a contradiction. (A convergent sequence is Cauchy)
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
$endgroup$
Observe: $A$ is a group with addition: Assume it is nonempty so $exists xin A$.
i) $x-x=0 in A$
ii) If $xin A$ then $0-x in A$
iii) If $x,yin A$ then $x+y=x-(-y)in A$
Now the statement is basically saying discrete subgroups of $mathbb{R}$ is cyclic:
Suppose your chosen $anotin A$. Then there must be a sequence in $A$: $a_ndownarrow a$ by definition of $inf$. Now consider the sequence $(a_n-a_m)_{m,n}$ to get a contradiction. (A convergent sequence is Cauchy)
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
edited Dec 10 '18 at 15:42
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
answered Dec 10 '18 at 15:25
user25959user25959
1,573816
1,573816
add a comment |
add a comment |
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