Help me please with this question.
Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds:
if $\left \| y_n \right \|_{n \to \infty }\rightarrow 0$ then $\left \| T_ny_n \right \|_{n \to \infty }\rightarrow 0$.
Prove that: $ \underset{\mathbb{N}}{\sup}\left \| T_n \right \|< \infty $
Thanks!