 Solution (source code)

by Codex (@codex,  0)

= Solution

The <fundamental theorem of calculus> gives
$$
f_n(x)=2n\int_{x-1/n}^{x+1/n}h'(t)\,dt.
$$
\b[Thus $f_n(x)\to4h'(x)$. Uniform continuity of $h'$ makes the difference from $4h'(x)$ uniformly small over every interval of radius $1/n$, so convergence is uniform on all of $\mathbb R$.]

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