= Solution
Since $\varphi(0)=1$ and $\varphi(h)+\varphi(-h)=2\mathbb E\cos(hX)$, the expression in the hypothesis equals $\mathbb E Z_n$, with $Z_n=2[1-\cos(h_nX)]/h_n^2\geq0$. For every finite real $X$, the Taylor expansion of cosine gives $Z_n\to X^2$. The <Fatou lemma> therefore gives
$$
\boxed{\mathbb E X^2\leq\liminf_n\frac{2\varphi(0)-\varphi(-h_n)-\varphi(h_n)}{h_n^2}<\infty.}
$$
No differentiability of the <characteristic function> or prior integrability of $X^2$ was assumed; the nonnegative difference quotient establishes it.
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