Let $X_n(t)$ and $X(t)$ are bounded stochastic processes ($|X_n(t)|<C$ and $|X(t)<C|$) such that for each $t\in[0,T]$ $$ X_n(t) \to X(t) \text{ as } n\to\infty \text{ almost surely }. $$
The question is following: Can we conclude $$ \int_0^T(X_n(t) - X(t))^2\, dt \to 0 \text{ as } n\to \infty \text{ almost surely }? $$
The claim is indeed a straight-forward application of the dominated convergence theorem. For fixed $\omega$ set $$f_n(t) := (X_n(t,\omega)-X(t,\omega))^2,$$ then by assumption $f_n(t) \to 0$ as $n \to \infty$ for each $t$ and moreover, $|f_n(t)| \leq C$ for some constant $C$ (not depending on $n$ or $t$). Applying the dominated convergence theorem proves the claim.
Note that it is important that the bound $C$ does not depend on $n$ or $t$; otherwise the claim does, in general, not hold.