= Solution
Write
$$
d_n(H)=\frac{\operatorname{ex}(n,H)}{\binom n2}.
$$
Let $G$ be an $H$-free <graph> on $n$ vertices with $\operatorname{ex}(n,H)$ edges, and choose a uniformly random set of $m<n$ vertices. Its <induced subgraph> is still $H$-free, and each edge survives with probability $\binom m2/\binom n2$. Hence
$$
\operatorname{ex}(m,H)\geq
\operatorname{ex}(n,H)\frac{\binom m2}{\binom n2},
$$
so $d_m(H)\geq d_n(H)$. Thus the normalized <extremal number> is nonincreasing and bounded below by zero, proving that
$$
\boxed{\ \lim_{n\to\infty}\frac{\operatorname{ex}(n,H)}{\binom n2}\ \text{exists}.\ }
$$
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