A problem about the convergence of convolution

Question

Let $G$ be a topological group. Let $f_{n}$ and $g_{n}$ be two sequences in $L^{p}(G)$ that are convergent to $f$ and $g$, respectively. Let $f * g \in L^{p}$. Is $f_{n} * g_{n}$ convergent to $f * g$? why? ($f_{n}*g_{n}\in L^{p}$ for all $n\in \mathbb N$).