Let $\{X_t\}$ be a square integrable martingale, $X^n_t\rightarrow X_t$ in $L^2(\Omega,\mathcal{F},P)$ for each $t$ (i.e., in the sense of mean square).
Do we have $[X^n,X^n]_t$ converges to $[X,X]_t$ for each t almost surely (or at least we can select a subsequence so that the convergence holds)?
Remark: We do not assume that $X$ has continuous path so the quadratic variation is not equal to the "predictable quadratic variation"(or "angel bracket process") which is induced by the Doob-Meyer decomposition.