To prove the Heine-Borel theorem you need to show that a compact set is both closed and bounded.
There is a proof of the theorem in the book The Elements of Real Analysis by Bartle. In the proof to show that a compact set K is closed, a specific open cover is used:
$$ G_{m} =\left \{ y \, \epsilon \, \mathbb{R}^{n}: ||y-x|| > 1/m \: \forall m \, \epsilon \, \mathbb{N} \right \}$$
where $$ x \, \epsilon \, \complement_{K} $$
I'm curious as to why we are allowed to use a specific open cover to prove a compact set is closed. My thought was that a general open cover should be used because compactness of a set is a property which is related to every open cover on that set.
Best Answer
The open cover that you mentioned are used to prove that if it is a compact set, then it is closed and bounded. Hence a particular open cover can be used.
The $G_{\alpha}$ in the proof in the converse direction should be viewed as a general open cover.