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" ?
Best Answer
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.