= Solution
{c}
The <cumulative distribution function> of the <uniform distribution> on $[0,1/n]$ is
$$
F_n(t)=\begin{cases}0,&t<0,\\nt,&0\leq t\leq1/n,\\1,&t>1/n.\end{cases}
$$
The constant zero has <cumulative distribution function> $F(t)=\mathbf1_{[0,\infty)}(t)$. Its <continuity points> are precisely $t\ne0$. For fixed $t<0$, $F_n(t)=0=F(t)$; for fixed $t>0$, $F_n(t)=1=F(t)$ eventually. This proves <convergence in distribution> to zero directly from the definition.
Also, $\mathbb P(0\leq U_n\leq1/n)=1$. For every $\varepsilon>0$ and $n>1/\varepsilon$, $\mathbb P(|U_n|>\varepsilon)=0$. Hence
$$
\boxed{U_n\longrightarrow0\text{ both in distribution and in probability}.}
$$
The <convergence in probability> conclusion holds for any realization of these <random variables> on a common <probability space>; no <independence of random variables> is needed.
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