Proving that for a martingale $X_n$ and stopping time $N$, $\lim_{n \to \infty} X_{N \wedge n} = X_N$

Question

Let $(X_n)_{n \geq 1}$ be a martingale w.r.t. $\mathcal{F}_n$. We define the stopped martingale as $X_{N \wedge n}$, where $N$ is a stopping time. It is well known that $(X_{N \wedge n})_{n \geq 1}$ itself forms a martingale. If we further assume that $N < \infty$ almost surely, it is also known that: $$ \lim_{n \to \infty} X_{N \wedge n} = X_N $$ Heuristically, this makes perfect sense. However, I have no idea how to prove this rigorously. A brief search online does not seem to yield any results.

Any help is appreciated.

Answer 1

$X_{N \wedge n}=X_N$ whenever $n \geq N$, hence $X_{N \wedge n} \to X_N$. You don't require any probability theory for this.