This is a book about writing proofs.

It is intended to develop that skill, not introduce you to a breadth of topics.

Well, there's too much of (modern) mathematics to touch everything - even everything important... One of the most important notions is that of compact sets, and it's not even mentioned. Neither are groups (well, they are mentioned, but not much else), etc.

From a quick skim of the ToC this looks very well thought-out. Looks like an excellent book!