Is this argument about sequential compactness wrong?

Question

In the 4th line of the first example of this page (it's a simple self-contained argument, I just doubt its validity), http://www.math.unl.edu/~s-bbockel1/929/node16.html they seem to be arguing that $g^{-1}(0)$ is compact (it is closed in a compact space), and therefore any sequence inside it must have a limit point.

But isn't that sequential compactness, rather than compactness? It is well-known that the two don't generally imply one another, if we're not in a metric space (What's going on with "compact implies sequentially compact"?).

Answer 1

"Every sequence has a limit point" is not the same as sequential compactness, which says that "every sequence has a convergent subsequence", but a variant of limit point compactness (or countable compactness) which does follow from compactness.