I'm reading the book Topics in Banach Space Theory by Albiac F. Kalton N. J. I got stuck at the proof of Pitt's theorem.
In the second paragraphs authors tries to prove ad absurdum that for weakly nuul sequence $\lim\limits_{n\to\infty}\Vert T(x_n)\Vert=0$. They say that without loss of generality one may suppose that $\{x_n\}_{n=1}^\infty$ is a weakly null sequence with $\Vert x_n\Vert=1$ and $\Vert T(x_n)\Vert>\delta$ for all $n\in\mathbb{N}$. I think they normalized original sequence $\{x_n\}_{n=1}^\infty$ and claims that it is also weakly null.
Why is weakly null sequence remains weakly null after normalization?
Another place I got stuck is the place where authors claims that passing to subsequence in $\{T(x_n)\}_{n=1}^\infty$ gives subsequence equivalent to the natural basis of $\ell_p$. And they also assume that after passing to subsequence $\{x_n\}_{n=1}^\infty$ remains to be equivalent to natural basis of $\ell_p$.
Why is $\{x_n\}_{n=1}^\infty$ remains to be equivalent to natural basis of $\ell_p$?
Thank you.