Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that $Lx=\lim_{n\rightarrow\infty}T_{n}x$, for all $x\in H$ (i.e. the $T_{n}$ converge pointwise to $L$).
How can I prove then, that $\left(T_{n}\right)_{n}$ converges with respect to the operator norm to $L$ (i.e. $\left(T_{n}\right)_{n}$ converges uniformly to $L$)?