Let $G$ be a locally compact abelian group, and let $f \in L^p(G), g \in L^q(G)$. I'm trying to prove that $f \ast g \in C_0(G)$. The book I'm reading (Rudin, Analysis on Groups) gives the following sketch of the proof:
Since $C_c(G)$ is dense in the $L^p$ spaces, there exist sequences $f_n \in L^p(G), g_n \in L^q(G)$ such that $||f - f_n||_p, ||g - g_n||_q \to 0$. It suffices then, to show that $f_n \ast g_n$ converges to $f \ast g$ uniformly. Since $f_n \ast g_n \in C_c(G)$ (we've already proved that $C_c(G)$ is closed under convolution), this will imply that $f_n \ast g_n \to f \ast g$ in the sup norm. Since $C_c(G)$ is dense in $C_0(G)$, the claim will follow.
But, I don't see why $f_n \ast g_n \to f \ast g$ uniformly. Rudin claims that this follows from the Holder inequality, but I'm having trouble making it work.
I'm not sure the following works. Please double check and let me know if I am wrong.
We have, for $x \in G$, \begin{align} |f_n*g_n(x)-f*g(x)| & = |f_n*g_n(x) - f*g_n(x) + f*g_n(x) - f*g(x)| \leq \\ & \leq |f_n*g_n(x) - f*g_n(x)| + |f*g_n(x) - f*g(x)| \leq \\ & \leq|(f_n - f)*g_n(x)| + |f*(g_n-g)(x)| \leq \\ & \leq \|f_n-f\|_p\|g_n\|_q + \|f\|_p\|g_n-g\|_q \leq \\ & \leq M\|f_n-f\|_p + \|f\|_p\|g_n-g\|_q \to 0. \end{align}
where we used the fact that $g_n$ is bounded in $L^q$ (since it is convergent) and that $f_n \to f$ in $L^p$ and $g_n \to q$ in $L^q$. Since the final line goes to zero uniformly in $x$ the convergence is uniform. Moreover we have used the fact that $L^p$ norm is invariant under translation.