Suppose we have functions $u_n\in C_c^\infty(\mathbb{R}^d)$ with support all lying in $B(0,R)$, and suppose $u_n\to 0 $ in $\mathcal{D}'(\mathbb{R}^n)$, i.e. for all $\eta\in C_c^\infty(\mathbb{R}^d)$, $$\int_{\mathbb{R}^n} u_n \eta\,\mathrm{d}x\xrightarrow{n\to\infty}0$$ then is it true that $\partial^\alpha u_n\to 0$ uniformly for all $\alpha$?

Suppose we have functions $u_n\in C_c^\infty(\mathbb{R}^d)$ with support all lying in $B(0,R)$, and suppose $u_n\to 0 $ in $\mathcal{D}'(\mathbb{R}^n)$, i.e. for all $\eta\in C_c^\infty(\mathbb{R}^d)$, $$\int_{\mathbb{R}^n} u_n \eta\,\mathrm{d}x\xrightarrow{n\to\infty}0$$ then is it true that $\partial^\alpha u_n\to 0$ uniformly for all $\alpha$?

The reason I'm asking this is because in Hörmander's The Analysis of Linear Partial Differential Operators I, in the paragraphs immediately above Definition 4.2.1, we have the following statement:

Let $u\in\mathcal{D}'(\mathbb{R}^n)$ have compact support, and then recall that $u*\phi\in C_c^\infty(\mathbb{R}^n)$ for all $\phi\in C_c^\infty(\mathbb{R}^n)$. Then in fact, the map $$\phi\mapsto u*\phi,\quad C_c^\infty(\mathbb{R}^n)\to C_c^\infty(\mathbb{R}^n)$$ is continuous.

However, I'm struggling to prove this fact.