The textbook I'm using has a somewhat confusing proof of one of the directions of Heine-Borel's Theorem, namely the one that states that a closed and bounded subset of $\mathbb R^n$ is compact. It uses sequences and "Changing the centers" to 'move' the balls to $\mathbb Q^n$ and then constructs a sequence which leads to a contradiction, the latter of which I'm confused about.

Could someone give me a more coherent proof in the same style or point me in the direction of a fairly understandable proof?