The definition of a bounded set $S$ in a metric space $X$, is simply that $$S\subset B_r(x) $$ for some $r>0$ and $x\in X$. But I've seen a few proofs where they use the fact that a set is bounded to say that it must be contained in some closed ball. I can't think of an example where this isnt true, but I wasn't sure if the two definitions are equivalent. Can someone give a proof? Thanks!