Let $\displaystyle g(x):\mathbb{R} \to \mathbb{R}$ be a function such that $\displaystyle \int_{-\infty}^\infty g(x) dx =1$. Show that for any continuous function $\displaystyle f:\mathbb{R} \to\mathbb{R}$,we ahve the sequence $\displaystyle g_n(x)=n\int_{-\infty}^\infty g(n(x-y))f(y)dx $converges uniformly on the compact sets of $\displaystyle \mathbb{R}$ to f(x)