If ${\int}_{\mathbb{R}}|f(x)|dx$ and ${\int}_{\mathbb{R}}|g(x)|dx$ are bounded, show that

${\int}_{\mathbb{R}}|(f\star g)(x)|dx\le {\int}_{\mathbb{R}}|f(x)|dx\cdot {\int}_{\mathbb{R}}|g(x)|dx$

From the definition we have $(f\star g)(x)={\int}_{\mathbb{R}}f(x-\tau )g(\tau )\phantom{\rule{thinmathspace}{0ex}}d\tau $. I guess I need to do work on this right side, but I don't see how to simplify to get the desired inequality

${\int}_{\mathbb{R}}|(f\star g)(x)|dx\le {\int}_{\mathbb{R}}|f(x)|dx\cdot {\int}_{\mathbb{R}}|g(x)|dx$

From the definition we have $(f\star g)(x)={\int}_{\mathbb{R}}f(x-\tau )g(\tau )\phantom{\rule{thinmathspace}{0ex}}d\tau $. I guess I need to do work on this right side, but I don't see how to simplify to get the desired inequality