= Solution
Apply part (b) simultaneously at every rational $t$; the intersection of these probability-one events still has probability one. On that event, monotonicity and continuity extend the convergence from the rationals to every real $t$.
More directly, fix $\varepsilon>0$. Since $F$ is a continuous <cumulative distribution function>, choose rational points
$$
t_0<t_1<\cdots<t_k
$$
so that the two tails have $F(t_0)<\varepsilon$ and $1-F(t_k)<\varepsilon$, while every increment $F(t_i)-F(t_{i-1})<\varepsilon$. Part (b) makes $|F_m^*(t_i)-F(t_i)|<\varepsilon$ at all these finitely many points for all sufficiently large $m$. If $t_{i-1}\leq t\leq t_i$, monotonicity gives
$$
F_m^*(t_{i-1})\leq F_m^*(t)\leq F_m^*(t_i),
$$
so $|F_m^*(t)-F(t)|<2\varepsilon$; the same estimate follows in the tails. This proves the <uniform convergence of distribution functions to a continuous limit>:
$$
\sup_{t\in\mathbb R}|F_m^*(t)-F(t)|\longrightarrow0
$$
almost surely.
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