The proof is a bit long, so I can't type/paste all of them here, my question is simply about last few sentences in proof of 8.3-5 (b), like in the image below:
My question is how we have $v \in X$, then $y \in T_\lambda(X)$, thus contradict the hypothesis.
$v$ is defined to be the limit of some sequence in $X$. Indeed, the $x_{n_k}$ were in $X$ by assumption (see the proof of part (a)), and the $z_{n_k}$ are in the null-space of $T_\lambda$, which is a subspace of $X$. Since $X$ is closed (in itself), the limit $v$ must also be in $X$.