Convergent Sequence in Test Function Space multiplied by Smooth Function

Theorem
Let $\alpha \in \map {\CC^\infty} {\R^d}$ be a smooth real multivariable function.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ a sequence in the test function space.

Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ converges in the test function space:


 * $\phi_n \stackrel \DD {\longrightarrow} \phi$

Then $\sequence {\alpha \phi_n}_{n \mathop \in \N}$ converges in the test function space.


 * $\alpha \phi_n \stackrel \DD {\longrightarrow} \alpha \phi$