Convolution of an $L^1$ function and a function that tends to $0$ results in a function that tends to $0$

Question

I'm trying to solve the following problem in review for a test, but have only partly succeeded:

Let $K \in L^1(\mathbb{R})$ and $f$ be a bounded, measurable function on $\mathbb{R}$, with $\lim_{x\to\infty} f(x) = 0$. Let $F(x) = \int_{\mathbb{R}} K(x-t)f(t)\mathrm dt$. Prove that $F(x)$ is finite for every $x$ and $\lim_{x\to\infty} F(x) = 0$.

Using Hölder's inequality it was straightfoward to show $F(x)$ finite for any $x$. Any suggestions on how one can show that $F(x) \to 0$ as $x \to \infty$?

Answer 1

Hint: rewrite the integral as $$F(x)=\int_{\mathbb R}K(s)f(x-s)\mathrm ds,$$ then use a dominated convergence argument.