Mixing
Compact set always contains its sup and inf
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I am sorry if my question might seem minor, but I just want to clarify this point for myself.
So in other thread on mathstackexchange, it is stated and proved that compact sets always contain their inf and sup. The explanation is straightforward, using Heine-Borel characterization, since compact set is bounded, hence it always has sup and inf. On the other hand, since compact sets are closed, then since by properties of sup and inf we can always devise a sequence of points in the set converging to sup/inf, hence sup/inf are in the set.
My problem with the above proof is the last part using closedness. The argument asserts that inf and sup belong to the compact set, because closed sets contain there limit points, but what if sup and inf are not limit points of the set? This could easily happen, if the sequence that we devise consists only of points equal to sup and inf.
Eg: Consider the set $[-2,2] cup {3}$, which is compact. But its sup is not a limit point.
Does the argument break down?
real-analysis general-topology compactness supremum-and-infimum
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
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add a comment |
$begingroup$
I am sorry if my question might seem minor, but I just want to clarify this point for myself.
So in other thread on mathstackexchange, it is stated and proved that compact sets always contain their inf and sup. The explanation is straightforward, using Heine-Borel characterization, since compact set is bounded, hence it always has sup and inf. On the other hand, since compact sets are closed, then since by properties of sup and inf we can always devise a sequence of points in the set converging to sup/inf, hence sup/inf are in the set.
My problem with the above proof is the last part using closedness. The argument asserts that inf and sup belong to the compact set, because closed sets contain there limit points, but what if sup and inf are not limit points of the set? This could easily happen, if the sequence that we devise consists only of points equal to sup and inf.
Eg: Consider the set $[-2,2] cup {3}$, which is compact. But its sup is not a limit point.
Does the argument break down?
real-analysis general-topology compactness supremum-and-infimum
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
$endgroup$
1
$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16
add a comment |
$begingroup$
I am sorry if my question might seem minor, but I just want to clarify this point for myself.
So in other thread on mathstackexchange, it is stated and proved that compact sets always contain their inf and sup. The explanation is straightforward, using Heine-Borel characterization, since compact set is bounded, hence it always has sup and inf. On the other hand, since compact sets are closed, then since by properties of sup and inf we can always devise a sequence of points in the set converging to sup/inf, hence sup/inf are in the set.
My problem with the above proof is the last part using closedness. The argument asserts that inf and sup belong to the compact set, because closed sets contain there limit points, but what if sup and inf are not limit points of the set? This could easily happen, if the sequence that we devise consists only of points equal to sup and inf.
Eg: Consider the set $[-2,2] cup {3}$, which is compact. But its sup is not a limit point.
Does the argument break down?
real-analysis general-topology compactness supremum-and-infimum
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
$endgroup$
I am sorry if my question might seem minor, but I just want to clarify this point for myself.
So in other thread on mathstackexchange, it is stated and proved that compact sets always contain their inf and sup. The explanation is straightforward, using Heine-Borel characterization, since compact set is bounded, hence it always has sup and inf. On the other hand, since compact sets are closed, then since by properties of sup and inf we can always devise a sequence of points in the set converging to sup/inf, hence sup/inf are in the set.
My problem with the above proof is the last part using closedness. The argument asserts that inf and sup belong to the compact set, because closed sets contain there limit points, but what if sup and inf are not limit points of the set? This could easily happen, if the sequence that we devise consists only of points equal to sup and inf.
Eg: Consider the set $[-2,2] cup {3}$, which is compact. But its sup is not a limit point.
Does the argument break down?
real-analysis general-topology compactness supremum-and-infimum
real-analysis general-topology compactness supremum-and-infimum
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
edited Jan 15 at 21:17
Renat Sergazinov
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
asked Jan 15 at 17:56
Renat SergazinovRenat Sergazinov
346
346
1
$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16
add a comment |
1
$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16
1
1
$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16
add a comment |
1 Answer
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A proof does not necessarily require limit point argument. Let $K$ be a compact subset of $Bbb R$ and $M=sup K$. Then by the definition of the supremum, for each $ninBbb N$, there exists $x_nin K$ such that $M-frac{1}{n}<x_nle M$. (Note that if $M$ is an isolated point of $K$, then $x_n=M$ for all sufficiently large $n$.) It easy easy to see that $limlimits_{nto infty} x_n =M$. Since $K$ is closed, it follows $Min K$. Similarly, $m=inf K$ also belongs to $K$. Thus $K$ has both maximum and minimum.
answered Jan 15 at 18:24
SongSong
14.1k1633
$endgroup$
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
add a comment |
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$begingroup$
A proof does not necessarily require limit point argument. Let $K$ be a compact subset of $Bbb R$ and $M=sup K$. Then by the definition of the supremum, for each $ninBbb N$, there exists $x_nin K$ such that $M-frac{1}{n}<x_nle M$. (Note that if $M$ is an isolated point of $K$, then $x_n=M$ for all sufficiently large $n$.) It easy easy to see that $limlimits_{nto infty} x_n =M$. Since $K$ is closed, it follows $Min K$. Similarly, $m=inf K$ also belongs to $K$. Thus $K$ has both maximum and minimum.
answered Jan 15 at 18:24
SongSong
14.1k1633
$endgroup$
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
add a comment |
$begingroup$
A proof does not necessarily require limit point argument. Let $K$ be a compact subset of $Bbb R$ and $M=sup K$. Then by the definition of the supremum, for each $ninBbb N$, there exists $x_nin K$ such that $M-frac{1}{n}<x_nle M$. (Note that if $M$ is an isolated point of $K$, then $x_n=M$ for all sufficiently large $n$.) It easy easy to see that $limlimits_{nto infty} x_n =M$. Since $K$ is closed, it follows $Min K$. Similarly, $m=inf K$ also belongs to $K$. Thus $K$ has both maximum and minimum.
answered Jan 15 at 18:24
SongSong
14.1k1633
$endgroup$
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
add a comment |
$begingroup$
A proof does not necessarily require limit point argument. Let $K$ be a compact subset of $Bbb R$ and $M=sup K$. Then by the definition of the supremum, for each $ninBbb N$, there exists $x_nin K$ such that $M-frac{1}{n}<x_nle M$. (Note that if $M$ is an isolated point of $K$, then $x_n=M$ for all sufficiently large $n$.) It easy easy to see that $limlimits_{nto infty} x_n =M$. Since $K$ is closed, it follows $Min K$. Similarly, $m=inf K$ also belongs to $K$. Thus $K$ has both maximum and minimum.
answered Jan 15 at 18:24
SongSong
14.1k1633
$endgroup$
A proof does not necessarily require limit point argument. Let $K$ be a compact subset of $Bbb R$ and $M=sup K$. Then by the definition of the supremum, for each $ninBbb N$, there exists $x_nin K$ such that $M-frac{1}{n}<x_nle M$. (Note that if $M$ is an isolated point of $K$, then $x_n=M$ for all sufficiently large $n$.) It easy easy to see that $limlimits_{nto infty} x_n =M$. Since $K$ is closed, it follows $Min K$. Similarly, $m=inf K$ also belongs to $K$. Thus $K$ has both maximum and minimum.
answered Jan 15 at 18:24
SongSong
14.1k1633
edited Jan 15 at 18:36
answered Jan 15 at 18:24
SongSong
14.1k1633
answered Jan 15 at 18:24
SongSong
14.1k1633
answered Jan 15 at 18:24
SongSong
14.1k1633
14.1k1633
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
add a comment |
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
I would guess that you need to consider cases in the proof then. First, consider M being an isolated point and argue that it is necessarily true that M is in the set. Second, consider M being a limit point and argue that it is in the set, becuase it is closed, i.e., it contains all of its limit points.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:21
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
$begingroup$
@RenatSergazinov Umm ... actually, I'm arguing that we do not need to think cases separately ... First, If $M$ is an isolated point, then $x_n$ should be equal to $M$ for all but finitely many $n$. If $Mnotin K$ ($Min K$ is not proved yet), then $x_n$ approaches to $M$ strictly from below. Two cases are dealt with automatically and simultaneously by the construction of $x_n$.
$endgroup$
– Song
Jan 15 at 21:23
add a comment |
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$begingroup$
Compact? Really?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 17:58
$begingroup$
You meant $[-2,2]cup{3}$. So forget "limit point"; it's easy to show that $sup Ainoverline A$ regardless.
$endgroup$
– David C. Ullrich
Jan 15 at 18:01
$begingroup$
I am sorry for my silly mistake.
$endgroup$
– Renat Sergazinov
Jan 15 at 21:16