= Solution
{c}
Use $X_n\sim\operatorname{Unif}[0,1/n]$ from part (a). We already proved <convergence in distribution> to zero. For any prescribed $p\in[0,1]$, put $c_n=p/n$. Then $c_n\to0$, and the <cumulative distribution function> satisfies
$$
\boxed{F_n(c_n)=F_n(p/n)=p\quad\text{for every }n.}
$$
This includes $p=0$ and $p=1$, since the <uniform distribution> assigns zero <probability> to each endpoint. The limit <cumulative distribution function> is discontinuous at zero, explaining why the conclusion of (ii) does not apply there.
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