If $T\in\mathcal{D}'(\mathbb{R}^n)$ and $\phi\in\mathcal{D}(\mathbb{R}^n)=C_c^\infty(\mathbb{R}^n)$ , then I want to show $\phi T$ is a distribution with compact support .
It suffices to show $\phi T=(\phi T)_f$ for some $f\in\mathcal{D}(\mathbb{R}^n)$ . Now for all $\psi\in C^\infty(\mathbb{R}^n)$ $$(\phi T)\psi=T(\phi\psi)$$ by definition . Afterwards how to show $\displaystyle T(\phi\psi)=\int_{\mathbb{R}^n}\phi(x)\psi(x)f(x)dx$ for some $f\in\mathcal{D}(\mathbb{R}^n)$ ? Any help is appreciated .
Take $f\in\mathcal{D}(\mathbb{R}^n)$ such that $f\equiv 1$ on a neighborhood of $\operatorname{supp}\phi.$