Let a sequence of test functions ${\phi_{m(x)}}, m = 1, 2, . . .$ converge in D to zero, and let $\psi(x)$ be an infinitely differentiable function. Show that $\psi\phi_{m(x)}$ also converges to zero.
Any help/hints would be greatly appreciated.
A sequence of ${\phi_{m(x)}}, m = 1, 2, . . .$ converges to $\phi \; $if - all ${\phi_{m(x)}}, m = 1, 2, . . .$ as well as $\phi$ vanishes outside the region. and $D^{k}{\phi_{m(x)}} \rightarrow D^{k}\phi$ uniformly over $R^{n}$ as $m$ goes to infinity for all multiindices $k$ and also we have $\psi \phi_{m}$ in D since $\psi$ is a function of $C^{\infty}$.