So I am working through some problems on my own and ran into one I have a question about. It is as follows. Let $(X\_1,\tau\_1)$ and $(X\_2,\tau\_2)$ be topological spaces, show that a set is closed iff whenever $\{x\_\gamma\}\_{\gamma\in\Gamma}\subseteq A$ with $x\_\gamma\rightarrow x\in X$ then $x\in A.$ I don't see how to use the hint that $\mathcal{U}(x)=\{U\in\tau\_1:x\in U\}$ form a directed set with $U\leq V$ implying $V\subseteq U$. Any help pointing me in the right direction would be appreciated.

Attempt: Suppose that whenever $\{x\_\gamma\}\_{\gamma\in\Gamma}\subseteq A$ with $x\_\gamma\rightarrow x\in X$ then $x\in A.$ For all $x\in X\_1$ s.t. $x\not\in A$ 
 (i.e. $x\in A^C$) we have that $x$ is not a limit point of $A$. Hence, $A^C$ is a neighborhood of $x$. Now we only need that $A^C$ is open, which is true since it is a neighborhood of all its points as $x$ was arbitrary. This should give us the first direction. I don't see how to get the other direction based using the provided hint.