Solution
= Solution
The defining relation gives $A_1^{(n)}=X_1^2-2M_1^{(n)}$. Applying $(r-s)^2\le2r^2+2s^2$ and the bound from part (a),
$$
\mathbb E(A_1^{(n)})^2\le2\mathbb E X_1^4+8\mathbb E(M_1^{(n)})^2\le2C^4+8C^4.
$$
Thus
$$
\boxed{\mathbb E(A_1^{(n)})^2\le10C^4,}
$$
uniformly in the dyadic mesh. This controls the approximations to <quadratic variation> without assuming that their limits already exist.