Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 22J / c / Solution

Since $\phi (0)=1$ and $\phi (h)+\phi (-h)=2\mathbb{E}\cos (hX)$, the expression in the hypothesis equals $\mathbb{E}{Z}_n$, with $Z_n=2[1-\cos (h_{n}X)]/h_n^2\ge 0$. For every finite real $X$, the Taylor expansion of cosine gives $Z_n\to X^2$. The Fatou lemma therefore givesNo differentiability of the characteristic function or prior integrability of $X^2$ was assumed; the nonnegative difference quotient establishes it.