= Solution
Fix $t$. Uniform continuity of the sample path on $[0,t]$ gives
$$
\delta_n(t)=\max_k|X_{t\wedge t_k^n}-X_{t\wedge t_{k-1}^n}|\longrightarrow0.
$$
Since $p<2$,
$$
A_t^{(n)}
=\sum_k|\Delta_k^nX|^2
\leq\delta_n(t)^{2-p}B_t^{(n)}.
$$
The assumed pathwise boundedness of $\sup_nB_t^{(n)}$ makes the right-hand side tend to zero almost surely. Part (b) also gives $A_t^{(n)}\to A_t$ in probability, so uniqueness of limits in probability yields $A_t=0$ almost surely. The vanishing-quadratic-variation result from part (a), after localization, makes $X$ identically zero. Consequently every $B_t^{(n)}$ is zero and $\sup_nB_t^{(n)}=0$ almost surely.
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