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 $\alpha$?