Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 23G / b / Solution

A compactly supported smooth function belongs to $H^s({\mathbb{R}}^n)$ for every real $s$. Part (a) therefore puts $u$ in $H^{s+4}$ for every $s$. Taking $s$ arbitrarily large and applying the Sobolev embedding theorem to every derivative shows that $u\in C^{\infty }({\mathbb{R}}^n)$.