Below is a lemma from Ikeda Watanabe's Stochastic Differential Equations that the space of right continuous square integrable martingales is complete under the metric specified below.
In the middle of the proof, I don't know why we get $E[|X_t^n - X_t|^2] \to 0$ as $n\to \infty$ from $\sup_{t \le T}|X_t^n - X_t|\to 0$ in probability.
Since we have for any natural number $T$, $E[|X_T^n - X_T^m|^2] \to 0$ as $n,m \to \infty$, and $|X_T^n - X_T^m|^2$ is a submartingale in time, we get $E[|X_t^n - X_t^m|^2] \to 0$ for any $t\le T$. Now by the completeness of $L^2$, we can get a $L^2$ limit $X_t$.
However, my argument does not use the uniform convergence in probability of $X^n \to X$.
How is this condition used in the proof here?
