We have a metric space (V,d). Proof that the following two properties are equivalent.
a) Every sequence $a_n \in V$ has a subsequence which converges to a element $x \in V$
b) For every subset $A\subset V$ with an infinite amount of elements, there exists a $x\in V$ such that for every $\delta>0$ the set $B(x;\delta) \cap A$ has infinite elements.
This exercise has two hints namely:
For a to b: choose a proper sequence $a_n \in A$
For b to a: take a sequence $a_n \in V$ and make a distinction if $\{a_n | n\in \mathbb{N}\}$ has infinite and finite elements
I have absolutely no clue on how to proceed. I find it difficult to already use the hints. So at the moment I'm stuck at finding/choosing a proper sequence.