= Solution
Let $\tau_m=\inf\{s:|X_s|\geq m\}$. The stopped process $X^{\tau_m}$ is bounded, so part (a) gives a continuous adapted quadratic variation $A^{[m]}$. These processes agree before the smaller stopping time, because their dyadic sums agree there and the limits are unique in probability. They therefore paste to a continuous adapted process $A$ with $A_{s\wedge\tau_m}=A_s^{[m]}$.
For fixed $t$ and $\varepsilon>0$,
$$
\mathbb P\left(\sup_{s\leq t}|A_s^{(n)}-A_s|>\varepsilon\right)
\leq\mathbb P(\tau_m\leq t)
+\mathbb P\left(\sup_{s\leq t}|A_s^{(n)}(X^{\tau_m})-A_s^{[m]}|>\varepsilon\right).
$$
The second term tends to zero by part (a), while continuity of $X$ on $[0,t]$ makes $\mathbb P(\tau_m\leq t)\to0$. This proves convergence <uniform convergence on compacts in probability>[uniformly on compact intervals in probability].
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