Giving a sequential proof of the Heine-Borel theorem

You don't need $E_j$ to be closed: just take its closure $F_j$ (or just any closed bounded interval containing it). Then $F_j$ is closed and bounded, and you can apply your argument to find a subsequence which converges in $F_j$ on each coordinate. Then, because $E$ is closed in all of $\mathbb{R}^n$, the coordinatewise limit of this subsequence must actually be in $E$.

Aim for a simpler result. Namely if $S$ is a sequence in E, it has a subsequence that will converge in the $i$th index.

Now we start with $S = S_0$. Construct a subsequence $S_1$ of that that converges in the first index. Then a subsequence $S_2$ of $S_1$ that converges in the second index (note, we only need the compactness of E and the previous lemma to show this). Keep going until we have a subsequence $S_n$ of $S_0\ldots S_{n-1}$ that converges in every index. Verify that that subsequence converges to a point.