Let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration and $N \in \mathbb{N}$, $\tau$ be a stopping time with $\tau \leq N$ and set $\tau_n := \lceil \tau 2^n \rceil 2^{-n}$ for every $n \in \mathbb{N}$.
Finally, let $(M_t)_{t \geq 0}$ be a right-continuous submartingale.
Show that $M_\tau$ is integrable with $\lim_{n\to\infty} \mathbb{E} [M_{\tau_n} 1_B] = \mathbb{E}[M_\tau 1_B]$ for all $B \in \mathcal{F}_\tau$.
I already know that $ A_n := \sum_{k=n}^\infty \left( \mathbb{E}[M_{\tau_k} \vert \mathcal{F}_{\tau_{k+1}} ] - M_{\tau_{k+1}} \right)$ are uniformly integrable. How to apply Doob's decomposition here?