= Solution
For an <effective Cartier divisor> $F$, the <divisor restriction exact sequence> gives
$$
0\to\mathcal O_X(kD-F)\to\mathcal O_X(kD)\to\mathcal O_F(kD)\to0.
$$
The scheme $F$ has dimension at most $n-1$, so the permitted scheme version of part (i) bounds $h^0(F,\mathcal O_F(kD))$ by $Ck^{n-1}$. For the infinitely many $k$ in the hypothesis,
$$
h^0(X,kD-F)\geq C'k^n-Ck^{n-1}>0
$$
once $k$ is sufficiently large. Thus \b[infinitely many such $k$ have a nonzero section of $kD-F$]. This dimension-drop argument is the <section subtraction lemma for big divisors>.
If “effective divisor” is interpreted as an effective <Weil divisor> on a normal variety, use its coherent divisor ideal instead. The quotient by that ideal is supported in dimension at most $n-1$, so the <polynomial bound for sections of a fixed divisor> gives the same conclusion. For $F=0$ the assertion is immediate.
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