Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 15A / d / Solution

Interpreting the displayed function aswe have ${\gamma }_n\ge 0$ and, after $u=nx$, ${\int }_{\mathbb{R}}{\gamma }_n=1$. Moreover its mass outside any fixed neighborhood of zero tends to zero. Therefore $({\gamma }_n)$ is an approximate identity, and for every test function $f$,Thus ${\gamma }_n\to \delta$ in the sense of distributions.