Originally Posted by

**javeiro** Hey,

I just moved from euclidean spaces to generic metric spaces, and I'm trying to avoid missing some basic concepts that links these spaces.

For example, in a metric space $\displaystyle (S,\rho) $, how can I define that some set is bounded? Can I say that $\displaystyle A \subseteq S$ is bounded if $\displaystyle \exists M \in \mathbb{R}$ s.t. $\displaystyle \rho (x,y)<M, \forall x,y \in A$ ? Assuming that S doesn't necessarily has a zero.

What I wanna show is that if $\displaystyle K \in S $ is compact, then every sequence in K has at least one convergent subsequence.

I can prove it in Rn using bolzano-weierstrass, but i'm unsure about what I can use in such a generic metric space.

Thanks in advance,

JP