= Solution
No <quadratic variation> argument is needed. For deterministic $u<v$, the <martingale> property and square integrability give
$$
\mathbb E(X_vX_u)=\mathbb E\bigl[X_u\mathbb E(X_v\mid\mathcal F_u)\bigr]=\mathbb E X_u^2.
$$
Consequently
$$
\mathbb E(X_v-X_u)^2=\mathbb E X_v^2-\mathbb E X_u^2.
$$
Apply this identity separately to each consecutive grid increment. The second moments telescope, giving
$$
\boxed{\mathbb E V_t^{n,|\cdot|^2}=\mathbb E X_{t_n}^2-\mathbb E X_0^2
\le\sup_{s\ge0}\mathbb E X_s^2-\mathbb E X_0^2<\infty.}
$$
This uniform bound follows directly from <martingale-difference orthogonality>, including the final interval specified by the ceiling.
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