= Solution
By the stated Cauchy property and completeness of $\mathcal M^2$, there is a square-integrable continuous martingale $M$ such that
$$
\sup_{t\geq0}\mathbb E|M_t^{(n)}-M_t|^2\longrightarrow0.
$$
Set $A_t=X_t^2-M_t$. This process is continuous and adapted, and
$$
\mathbb E\sup_{t\geq0}|A_t^{(n)}-A_t|^2
=\mathbb E\sup_{t\geq0}|M_t^{(n)}-M_t|^2
\leq4\sup_{t\geq0}\mathbb E|M_t^{(n)}-M_t|^2\longrightarrow0
$$
by the <Doob L2 maximal inequality>. The process $A$ is the <quadratic variation> $[X]$.
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