= Solution
For every $t$, subtraction of the definitions gives
$$
A_t^{(n)}-A_t^{(m)}=-2(M_t^{(n)}-M_t^{(m)}).
$$
The difference of the two supplied <continuous martingales> is a square-integrable martingale on $[0,1]$; for each fixed mesh its finite sums are bounded. For this <dyadic quadratic variation of a bounded continuous martingale>, the <Doob L2 maximal inequality> therefore gives
$$
\mathbb E\sup_{0\le t\le1}|A_t^{(n)}-A_t^{(m)}|^2\le16\mathbb E|M_1^{(n)}-M_1^{(m)}|^2\longrightarrow0.
$$
Thus \b[the dyadic approximations to quadratic variation are Cauchy for the expected squared uniform norm], as required. No monotonicity in time of the partially completed squared-increment sums is needed.
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