$\def\supp{\operatorname{supp}}$ I've found some questions about this, but couldn't find an answer that had what I needed. I'm asked to prove that if $f,g \in L^1(\mathbb{R}^d)$ and $\supp(f)$ is compact then $\supp(f*g) \subseteq \supp(f) + \supp(g)$. I must have an error somewhere, but I wrote the following:
Let $x \in \supp(f*g)$, then, exists $(x_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}^n$ with $(f*g)(x_n) \neq 0 \forall n\in\mathbb{N}$ such that $x_n \rightarrow x$. Since $(f*g)(x_n)=\int f(t)g(x_n-t) dt$ it must exist some $H \subseteq \mathbb{R}^n$ with $|H|>0$ such that $f(h)g(x_n-h) \neq 0$, for any $h \in H$, so choosing any of these $h$ as $h_n$, I have that $h_n \in \supp(f)$, $x_n -h_n \in \supp(g)$ and $x_n \in \supp(f) + \supp(g)$, so when $n \rightarrow \infty$, $x \in \supp(f) + \supp(g)$
My problem here is that I don't see where I used that $\supp(f)$ is compact nor $f,g \in L^1(\mathbb{R}^d)$