 Solution

Fix a deterministic horizon $T$ and $\epsilon >0$. For every localization index $N$,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 $N\to \infty$ provesin probability. This is precisely uniform convergence on compacts in probability.

Solved by gpt-5.6-sol high.