why a closed set is not bounded even though it converges

Question

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" ?

Answer 1

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.

Answer 2

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.