Clarify proof of theorem 6.6 Fredholm alternative in Haim Brezis book (Funtional Analysis)

Question

I'm currently reading proof of this theorem.

My question is in part b), it says that:

I have already wrote

But i'm not sure why $w_{n_k}-Tw_{n_k}$ corverges to $0$, which is implies from $(4)$. Can you clarify this? Thank you a lot.

Answer 1

The condition $(4)$ gives you the expression you wrote, $$ w_{n_k}-Tw_{n_k}=\frac{f_{n_k}}{\|u_{n_k}-v_{n_k}\|}. $$ The fact that $\{f_{n_k}\}$ converges guarantees that it is bounded. And the hypothesis is that the denominator goes to infinity, so the whole thing goes to zero.