 Solution

Fix $t$. Uniform continuity of the sample path on $[0,t]$ givesSince $p<2$,The assumed pathwise boundedness of ${\sup }_n{B}_t^{(n)}$ 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 ${\sup }_n{B}_t^{(n)}=0$ almost surely.