Prove that \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{2}(\mu)\) converges in \(\mathcal{L}^{2}\) if, and only if, \(\lim _{n, m ightarrow \infty} \int u_{n} u_{m} d \mu\) exists.

[ verify and use the identity \(\|u-w\|_{2}^{2}=\|u\|_{2}^{2}+\|w\|_{2}^{2}-2 \int u w d \mu\).]