The following is Ch4 Ex18(c) of Stein's Real Analysis / Measure Theory.
Exercise 4.18(c) Let $\mathcal H$ denote a Hilbert space and $\mathcal L(H)$ the vector space of all bounded linear operator on $\mathcal H$. Show that $\mathcal L(H)$ is complete in the metric $d$, where $d(T_1,T_2) = ||T_1 - T_2||$.
This is what I've done. ($\Vert\cdot \Vert$ implies $\Vert\cdot \Vert_{\mathcal H}$)
Let{$T_n$} be a Cauchy sequence in $\mathcal L(H)$; for every $\epsilon > 0$ there is $N_{\epsilon} \in \mathbb N$ such that $m,n \ge N_{\epsilon}$ implies $\Vert T_m - T_n \Vert < \epsilon$
$\Vert(T_m-T_n)f\Vert \le \Vert T_m-T_n\Vert \Vert f\Vert$ for every $f \in \mathcal H$, so $m,n \ge N_{\epsilon}$ implies $\Vert T_mf-T_nf \Vert \le \Vert f\Vert\epsilon$.
So, {$T_nf$} is a Cauchy sequence in $\mathcal H$. Then, there is $F \in \mathcal H$ such that $\lim_{n\to \infty} \Vert T_nf - F\Vert = 0$. Let $F = T(f)$.
-editted- If this is correct, it remains to show that $T$ is bounded, linear, and $\lim_{n\to \infty} T_n = T$. Except for the linearity, I have a difficulty solving these points.
Any help about this would be appreciated. Thank you.