Assume $\phi$ to be a nonnegative continuous function on the real line with compact support. Also assume that integral of $\phi$ over $\mathbb{R}$ is normalized to $1$. Let $\phi_e(x) = \frac{1}{e}\phi\left(\frac{x}{e}\right)$.
I succeed to prove that $\phi_e$ is an approximation to the identity & the convoluton of $\phi_e$ with $L^p$ function $g$ converges to $g$ in $L^p$ norm.
What I really need help with is the following. Prove or disprove when $p=\infty$. I thought my proof didn't work for $p = \infty$, so I tried all night to find counterexamples. But I failed. Can anyone give an answer or idea for my question?