Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 26G / b / iii / Solution

Continuity of $\phi$ at zero and $\phi (0)=1$ allow $u>0$ so small thatPointwise convergence and dominated convergence make the corresponding expression for ${\phi }_n$ less than $\epsilon$ for all sufficiently large $n$. Part (ii) then boundsThere are only finitely many remaining $n$, so enlarge $M\ge 2/u$ until each of their tails is also at most $\epsilon$. This proves uniform tightness.