Problem about convergence of operator norm and compactness of an operator. Problem-$1$ Given the sequence of continuous linear operators $T_n : l^2 \to l^2$ defined by $$T_n(x) = (0, 0, \ldots, x_{n+1}, x_{n+2}, \ldots)$$ for every $x \in l^2$. Then for every $x \neq 0$ in $l^2$ i want to check whether $\|T_n\|$ and $\|T_nx\|$ converge to $0$ or not?
Problem-$2$ Let the continuous operator $T: l^2 \to l^2$ defined by $$T_n(x) = (0, x_1, 0, x_3,  \ldots,)$$ for every $x = (x_1, x_2, \ldots) \in l^2$.  To find whether $T$ is compact or not?
According to me for $(1)$, $$\|T_nx\|^2 = \sum_{k = n+1}^{\infty} \|x_k\|^2$$ Now as $x \in l^2$, so there exist $N \in \mathbb{N}$ such that $$\sum_{k = N+1}^{\infty} \|x_k\|^2 < \epsilon$$ we conclude that $\|T_n(x)\| \to 0$. What about the others? am i correct?

$||T_n x|| \to 0$ for each $x$, but $||T_n e_{n+1}||=1$ so $||T_n|| \geq 1$ for each $n$. For 2) the answer is no:  $Te_1=e_2,Te_3=e_4,...$ so $\{T{e_{n+1}}\}$ has no convergent subsequence. Hence $T$ is not compact.
