Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$
Is my proof correct?
$\lim_{n\to\infty}\|Lu_n-Lu\|=\lim_{n\to\infty}\|L(u_n-u)\|\leq \lim_{n\to\infty}M||u_n-u||=0$
Therefore, $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$
Solution according to feedback:
the sequence $\{u_n\}$ converges implies $$\forall \epsilon>0, \exists N s.t. ||u-u_n||\leq \frac{\epsilon}{M} \forall n\geq N$$
Now,
$$||Lu_n-Lu||=||L(u_n-u)||\leq M||u_n-u||\leq M\frac{\epsilon}{M}=\epsilon \forall n\geq N$$
Therefore, $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$