Weyl's criterion for finding the essential spectrum. Necessary and sufficient?

Question

In this book, Lemma 2.16, it is stated:

However, in the book it is claimed that this is actually an if and only if criterion:

So in the first reference the Weyl sequence need not have a zero weak limit, and there is not an "if and only if" claim. I believe only the latter is correct. But am I missing something here?