Consider the following lemma:
In the fragment:
Write $\displaystyle X= \bigcup_{1}^{\infty} X_j$ where each $X_j$ is a compact subset of a ball $B$ as above.
why is he allowed to do that? If the $X_j$ were such that they were each inside a compact subset of a ball $B$ as above, I would understand. But this way, I don't see how this construction is possible. My question is: How is this construction performed?