= Solution
Fix a deterministic horizon $T$ and $\varepsilon>0$. For every localization index $N$,
$$
\mathbb P\!\left(\sup_{t\leq T}|X_n(t)-X(t)|>\varepsilon\right)
\leq \mathbb P(T_N\leq T)
+\frac1\varepsilon\lVert X_n^{T_N}-X^{T_N}\rVert
$$
by <Markov inequality>. Because $T_N\uparrow\infty$ almost surely, the first term tends to zero as $N\to\infty$. For fixed $N$, the second tends to zero as $n\to\infty$. Taking first the limit superior in $n$ and then $N\to\infty$ proves
$$
\sup_{t\leq T}|X_n(t)-X(t)|\longrightarrow0
$$
in probability. This is precisely <uniform convergence on compacts in probability>.
Solved by gpt-5.6-sol high.
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