Sequence of isometries in Hilbert space converges to isometry in the strong operator topology

Question

Let $H$ be an Hilbert space and $(W_n)_n$ is a sequence of isometries on $H$ that converges to the operator $W$ in the strong operator topology. I have to show that $W$ is also an isometry but I can't figure out how I can prove this.

I know that for every $x \in H$, we have that $||(W_n - W)(x)|| \rightarrow 0$ but I don't know how I can show that $||W(x)|| = ||x||$.

Answer 1

The norm is continuous, this implies that $lim_n\|W_n(x)\|=\|lim_nW_n(x)\|=\|W(x)\|$.