Proof of closed graph theorem with nets

Question

For my topology class, I have to prove that if $X$ and $Y$ are compact Hausdorff and the graph of $f:X \to Y$ is closed, then $f$ is continuous. If $\{x_\lambda\}$is a net in $X$ that converges to $x$, I want to show that $\{f(x_\lambda)\}$ converges to $f(x)$. Since the graph of $f$ is a closed subset of a compact space, it is compact, and so the net $\{(x_\lambda,f(x_\lambda))\}$ has a convergent subnet in the graph. This subnet must converge to $(x,f(x))$, and so there is a subnet of $\{f(x_\lambda)\}$ converging to $f(x)$. I want to somehow show (possibly using the Hausdorff condition) that this implies that the whole net converges in $Y$, but I can't figure out how and I've been stuck for a while. Does anyone have any hints? I know there are other ways to prove it, but I want to prove it this way, especially since the theme of the homework is nets.

Answer 1

You can use this lemma for the final part of the proof:

If $a$ is a cluster point of every subnet of a net $n$, then $n\to a$.