How to know if a space has a convergent subsequence?

Question

So this is something I have been struggling with lately... how do we generally know that a space/set has a subsequence that converges?

The current one I am struggling with is the space of sequences of real numbers that converge...

I have seen proofs of convergence (or proving something is Cauchy) usually start with something along the lines of "Let $\{x_n\} \in S$ be a convergent subsequence"... but how do we actually know if a set has one?

Thanks for helping out with this!

Answer 1

Compactness of $S$ is the key.

Answer 2

In any non-empty space, X, the sequence $a_n= a$, where a is any member of X, trivially converges to a.