Let $(\Omega,\mathcal{F},(\mathcal{F_n})_n,P)$ be a filtered probability space, $p \in (1,\infty)$, and $M_n$ an $\mathcal{F_n}$ martingale uniformly bounded in $L^p$.
I used this fact to show that $(M_n)_n$ is uniformly integrable and that it converges in $L^1$ to some integrable $M_{\infty}$ and $(M_n)_{\mathbb{N} \cup \infty}$ is a martingale.
How can I show that $|M_n-M_\infty|^p$ is uniformly integrable which will imply $M_n \to M_\infty$ in $L^p$?
This follows from what was done here: we use Doob's inequality to prove that for any $N$, $$\mathbb E\left[\max_{1\leqslant n\leqslant N}\left|M_n\right|^p\right] \leqslant \left(\frac p{p-1}\right)^p \mathbb E\left[\left|M_N\right|^p\right] \leqslant \left(\frac p{p-1}\right)^p \sup_{i\geqslant 1} \mathbb E\left[\left|M_i\right|^p\right] $$ and this proves, by monotone convergence, that $\max_{n\geqslant 1}\left|M_n\right|^p$ is integrable, which implies uniform integrability.
