Let \(1 \leqslant p<\infty\) and \(u, u_{n} \in \mathcal{L}^{p}(\mu)\) such that \(\sum_{n=1}^{\infty}\left\|u-u_{n}ight\|_{p}<\infty\). Show that almost everywhere \(\lim _{n ightarrow \infty} u_{n}(x)=u(x)\).

[ mimic the proof of the Riesz-Fischer theorem using \(\sum_{n}\left(u_{n+1}-u_{n}ight)\).]