= Solution
The identity map
$$
\operatorname{id}:
(\widehat{\mathcal P}_n,d_\infty)
\longrightarrow
(\widehat{\mathcal P}_n,d_1)
$$
is a continuous bijection by $d_1\le d_\infty$. Its domain is compact by part (i), and its codomain is a <Hausdorff space> because every metric space is Hausdorff. The <compact-to-Hausdorff continuous bijection theorem> therefore makes the identity a homeomorphism. Consequently
$$
\boxed{d_1\text{ and }d_\infty\text{ induce the same topology on }\widehat{\mathcal P}_n}.
$$
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