Let $S$ be a set of complex numbers. I would like to prove the following are equivalent:
1) every sequence of elements of $S$ has a point of accumulation in $S$;
2) every infinite subset of $S$ has a point of accumulation in S
3) every sequence of elements of $S$ has a convergent subsequence whose limit is in $S$
PROOF. 1)$\rightarrow$ 2) and 3)$\rightarrow$ 1) are trivial. How to prove 2)$\rightarrow$ 3)?