How to prove the Closed map lemma

The theorem is true in topological spaces in general, not just spaces in which sequences determine the topology (like metric spaces, for instance), so it can be proved without any use of sequences. In the general setting it's nets and filters that generalize sequences, but you don't need them here. If you want to learn more about them, a very good starting point is these notes by Pete L. Clark, who also posts here.

Suppose that $H\subseteq X$ is closed, and let $K=f[H]\subseteq Y$. Let $\mathscr{U}$ be a family of open subsets of $Y$ such that $K\subseteq\bigcup\mathscr{U}$; that is, $\mathscr{U}$ is an open cover of $K$ in $Y$. Let $\mathscr{V}=\{f^{-1}[U]:U\in\mathscr{U}\}$; since $f$ is continuous, $\mathscr{V}$ is an open cover of $H$ in $X$. But $H$, being a closed subset of the compact space $X$, is compact, so some finite subset $\{V_1,\dots,V_n\}$ of $\mathscr{V}$ covers $H$. For $k=1,\dots,n$ there is $U_k\in\mathscr{U}$ such that $V_k=f^{-1}[U_k]$, and it's clear that $\{U_1,\dots,U_n\}$ covers $K$. Thus, every open cover of $K$ has a finite subcover, and $K$ is compact. Now just use (or prove) the result that every compact subset of a Hausdorff space is closed.

I think that in Brian's proof he proved in the process that the continuous image of a compact set is compact.

Now in your proof above, you used sequences but you can prove it without using any of these The advantage of this is that your proof will almost generalise immediately to a general topological space. Here's how you use up all your hypotheses:

Let $B$ be a closed subset of our compact space $X$. Since a closed subset of a compact space is compact, we have that $B$ is a compact subset of $X$. Now by assumption $f$ is a continuous function so that $f(B)$ is compact. But then $Y$ is a Hausdorff space so that compact subsets of $Y$ are closed. It follows that $f(B)$ is closed and hence that $f$ is a closed map.