$f, g ∈L^1$, and $g$ is bounded. $f \ast g$ is the convolution, i.e. $f\ast g(x)=\int_{R}{f(x-y)\ g(y)dy}$.
Prove that $f \ast g→0$, when $|x|→+∞$.
This is an exercise from the Real Analysis of Stein.
I've proved that the convolution is uniformly continuous, but I don't know what to do next.
I also try to imitate the proof of the Riemann-Lebsgue Theorem, but fail, as the function is no longer periodic.
Any hint will be appreciated!!
Hint: use what's said in the comment part. You can find a series of continuous functions with compact support that approximate the f in $L^1\ $ norm. And the situation when f is a continuous functions with compact support is trivial.