= Solution
For each fixed $t\in[0,1]$, the variables $\mathbf1_{\{X_k\leq t\}}$ are <independent and identically distributed random variables> with <Bernoulli distribution> of parameter $t$. Their <expected value> is $t$, and the <strong law of large numbers> gives $F_n(t)\to t$ <almost surely>.
For a finite set $F$, intersect these probability-one events over its finitely many elements. On the resulting single event, every one of the finitely many errors tends to zero, so their maximum tends to zero:
$$
\boxed{\sup_{t\in F}|F_n(t)-t|\longrightarrow0\quad\text{almost surely}.}
$$
If $F$ is empty, take the displayed maximum to be zero. The important point is simultaneous convergence on one event; pointwise convergence at an uncountable collection of arguments would not alone imply the uniform claim in part (ii).
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