Why is the one-point compactification of the rationals sequentially compact?

Question

The problem is: Let $\Bbb Q^* = \Bbb Q \cup \{\infty\}$ be the one-point compactification of $\Bbb Q$. Is this sequentially compact?

A solution makes reference to $\Bbb Q^*$ being a sequentially compact space but without a proof, but wouldn't any sequence of rationals converging to an irrational have no convergent subsequence in $\Bbb Q^*$? How does this fail to be a counterexample?

Note that in this course we haven't studied first countable sets or anything to do with that. And thank you!

Answer 1

Hint: What is the topology on $\mathbb{Q}^*$? What does a neighborhood of $\infty$ look like? It's not at all what it would be in the case of $\mathbb{R}$.

Further Hint: $\mathbb{Q}$ is not locally compact, so the one point compactification is not Hausdorff.