= Solution
Choose a <very ample divisor> $A$ such that $A-D$ has a nonzero <global section>; this is possible by taking a sufficiently high ample twist. Since $X$ is integral, multiplication by its $m$th power injects $H^0(X,mD)$ into $H^0(X,mA)$. By <Serre vanishing> and the <Hilbert polynomial>, $h^0(X,mA)=O(m^n)$. Consequently
$$
\boxed{h^0(X,mD)\leq C m^n\quad(m\gg0).}
$$
This is the <polynomial bound for sections of a fixed divisor>. The same proof works for a <coherent sheaf> of support dimension $d$, by choosing the multiplying section to avoid its <associated points>; the bound is then $O(m^d)$.
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