Let $E$ be a topological vector space and $A \subset E$. How can I prove that $A$ is relatively countable compact if, and only if, every sequence in $A$ has a subnet that converges in $\bar A$.
I know the result just for countable compactness and I've managed to prove the result just for compactness, but in that proof I've constructed a new net in $A$ based on a net in $\bar A$ and a local base for $E$ and I don't know how to get in that way a sequence instead of a net.
Any ideas?
PD: The forward direction is easy to show. Is the reciprocal the one which I cannot prove.