Here's my proof in my own words, does it stack up?
Showing $A$ is complete implies $A$ is closed. Let $(x_n)$ be a convergent sequence in $A$. $A$ is complete $\implies (x_n) \to p \in A$. Hence $A$ is closed.
Showing $A$ is closed $\implies$ $A$ is complete. Let $(x_n)$ be a convergent sequence in $A$. $A$ is closed $\implies (x_n) \to p \in A$. As every convergent sequence is a Cauchy sequence, $(x_n)$ is a Cauchy sequence in $A$ that converges to $p \in A$ and hence $A$ is complete.
That sound ok?
As an aside, it seems to me that the definition of completeness and closed are basically identical, why are there two definitions for the same thing? Am I missing something here?