open subset of $P$, so that its closure $\overline U$ is compact. Consider the subset $X$ of $\mathscr E'^n$ consisting of distributions $\xi$ of degree at most $n$ with support in $U$ and such that $|\xi|'_{\overline U,n}{\leqslant}\gamma$. Then the seminorm $|\cdot|'_{\overline U,n}$ is nondegenerate on $X$, and within $X$ it acts as a norm that induces the topology that $X$ inherits from $\mathscr E'^n$, that is, the weak\* topology with respect to $C^\infty(P)$. In particular, the sequential closure of subsets of $X$ coincides with their closure.

By means of a partition of unity, we may assume at the outset that both $\mu$ and $\eta$ are compactly supported.

Let us explain how we will apply Corollary \[cor:useful\]. Let $n>0$ be the degree of $\eta$. Set $V=\mathscr E'$, $C=\mathscr M_0$, $p=\mu$, and $v=\eta$, and from the corollary we will get $c(t)=\mu_t$. Fix a bounded, open subset $U$ of $P$ that contains the supports of $\mu$ and $\eta$, and let $\gamma=2|\eta|'_{\overline U,n}$ Let the set $X\subset \mathscr E'^n$ be the set of distributions $\xi$ of order at most $n$ whose supports are contained in $U$ and such that $|\xi|'_{\overline U,n}{\leqslant}2|\eta|'_{\overline U,n}$. From Lemma \[lem:boundednormability\], we know that the topology on $X$ is induced 