Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 23H / c / Solution

Fix $R>0$ and choose ${\chi }_R\in C_c^{\infty }((-R-1,R+1))$ with ${\chi }_R=1$ on $[-R,R]$. The sequence $({\chi }_R{f}_n)$ is bounded in $H_0^1((-R-1,R+1))$. By part (a), each subsequence has a further subsequence converging strongly in $L^2$ on this bounded interval. The original weak convergence in $L^2(\mathbb{R})$ forces every such strong limit to be zero. It follows that the whole sequence satisfiesotherwise a subsequence bounded away from zero would contradict the preceding compactness argument.

The pointwise hypothesis gives a uniform tail estimate:whose right-hand side tends to zero as $R\to \infty$, independently of $n$. Given $\epsilon >0$, first choose $R$ so that this tail is below $\epsilon /2$, and then choose $n$ so that the integral on $[-R,R]$ is below $\epsilon /2$. Therefore