Def. 1. $\phi:\mathbb{R}^n\to \mathbb{R}$ is called test fuction if $\phi$ is infinitely differentiable ($\phi \in C^{\infty}(\mathbb{R}^n)$) and $\phi$ has compact support (ie the closure of the set $\{ x\in \mathbb{R}^n\,:\,\phi(x)\neq 0\}$ is a compact subset of $\mathbb{R}^n$).
Def.2. Let $\{\phi_m\}_m$ be a sequence of test functions. $\{\phi_m\}_m$ is called a null sequence if:
- There is a compact set $K\subset \mathbb{R}^n$ containing the supports of all $\phi_m$, $$\bigcup_{m\in \mathbb{N}}\operatorname{supp}(\phi_m)\subset K. $$
- For each multi-index $k=(k_1,\dots,k_n)$ $$\lim_{m\to +\infty}\max_{x\in \mathbb{R}^n}|D^k\phi_m(x)|=0$$ where $D^k\phi_m$ denotes the partial derivative of $\phi_m$, $$D^k\phi_m=\frac{\partial^{k_1+\cdots+k_n}}{{\partial x_1}^{k_1}{\partial x_2}^{k_2}\cdots {\partial x_n}^{k_n}}\phi_m.$$
Note that 2. is equivalent saying that $\{\phi_m\}$ converges uniformly to $0$ in $\mathbb{R}^n$, and so does the sequence $\{D^k\phi_m\}$ for every $k$.
What I have to show is the following
Prop. Let $\{\phi_m\}_m$ a null sequence and $a(x)\in C^{\infty}(\mathbb{R}^n)$. Then $\{a\phi_m\}_m$ is a null sequence.
proof
For every $m\in \mathbb{N}$ the product function $a\phi_m$ is smooth since $a,\phi_m$ are smooth and also $\operatorname{supp}(a\phi_m)\subset \operatorname{supp}(\phi_m)$. Then $\{a\phi_m\}_m$ is a sequence of test functions. Property 1 from Def.2. for $\{a\phi_m\}_m$ easily follow since $$\bigcup_{m\in \mathbb{N}}\operatorname{supp}(a\phi_m)\subset \bigcup_{m\in \mathbb{N}}\operatorname{supp}(\phi_m).$$ It remains to show that $$\lim_{m\to +\infty}\max_{x\in \mathbb{R}^n}|D^k\,a\phi_m(x)|=0$$ for every $k$.
I really don't know how to proceed in order to prove the last statement. Any hint would be really appreciated. Thanks in advance.
$D^k a\phi_m$ can be written as a linear combination of terms of the form $D^c a D^d \phi_m$ where $c$ and $d$ are multi-indices. $|D^c a|$ will attain a maximum value since it vanishes outside of $K$ so you can estimate that by a constant and then just use the fact $\{\phi_m\}$ is a null sequence.