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I looked at this question Does closed imply bounded?

according to the definition of closed set

A set $S$ in $\mathbb{R}^m$ is closed if, whenever $\{\mathbf{x}_n\}_{n=1}^{\infty}$ is convergent sequence completely contained in S, its limit is also contained in $S$.

Boundedness property of the convergent sequence says that every convergent sequence is bounded. Does not it mean that by default closed sets are bounded ?. Or is it just that the Boundedness property exclude the case of "converging to infinity" ?

José Carlos Santos
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Shew
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  • No, for example $\mathbb{R}^m$ satisfies this criterion (in a trivial manner). Note that you assume that $x_n$ is convergent in the first place and so you can't say anything about non-convergence sequences. One more thing, the phrase "convergent sequence" does not include sequences which converge to infinity, if a sequence convergence to infinity then it's diverge. – Yanko Jun 29 '18 at 20:56
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    Saying that a closed set converges doesn't make sense. Sequences can converge. – egreg Jun 29 '18 at 21:07
  • @egreg, does that mean a closed set is one which has at least one converging sequence in it ? – Shew Jun 29 '18 at 21:10
  • @Shew Not at all; every non empty set has a convergent sequence in it (constant). – egreg Jun 29 '18 at 21:28
  • Convergence does not apply to closed sets – William Elliot Jun 30 '18 at 02:19

2 Answers2

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No. It is indeed true that every convergent sequence is bounded, but it doesn't follow from that that every closed set is bounded. For instance, every metric space is a closed subset of itself, but, in general, metric spaces are not bounded.

José Carlos Santos
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  • i am trying to understand why ?. according to the definition of closed set it is a converging sequence. my question is what is the "extra property" it got that does not make it bounded. it seems it is not the "converging to infinity" case. – Shew Jun 29 '18 at 21:06
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    This has nothing to do with converging to infinity. And the definition of closed set is not that it is a closed sequence. That property that you mentioned says that a set $C$ is closed if and only if when a sequence of elements of $C$ converges, then its limit belongs to $C$. But t does not say that the elements of $C$ form a convergent sequence. – José Carlos Santos Jun 29 '18 at 21:26
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A closed set could be bounded or unbounded and a bounded set could be closed or not closed.

For example the set of integers is closed and unbounded while the interval $[0,1]$ is closed and bounded.

Convergent sequences in a set being bounded does not mean the set itself is bounded, for example in the set of natural numbers every convergent sequence is constant therefore it is bounded, but the set of natural numbers is unbounded.

Mohammad Riazi-Kermani
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