Say we have a martingale $X$ and a stopping time $T$.
Instead of directly studying the stopped process $X_T$, many proofs employ a trick, namely, one considers the bounded stopping time $T\land n$ and instead studies $X_{T\land n}$ and then letting $n \to \infty$ at some point.
I'm trying to prove that $$ \lim_n X_{T \land n} = X_T \mathbf{1}_{\{T < \infty\}}.$$
Intuitively it makes sense but I'm stuck trying to prove it.
Edit Update: Like saz noted in the comments below the above limit is not true in general if $P(T = \infty) > 0$.
I consider this solved as it cleared up my misunderstanding.