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A Cartier divisor $D$ on an integral projective variety is big exactly when, for every ample divisor $A$, some positive integer $j$ satisfies $jD\sim A+E$ with $E$ effective. To prove the forward direction, subtract an effective high multiple of $A$ using the section subtraction lemma for big divisors, then add an effective representative of the remaining multiple. For the reverse direction, multiply sections of $qA$ by the section of $qE$ and use the ample Hilbert polynomial.

Fix any ample divisor $A$. Choose $q\gg 0$ so that both $qA$ and $(q-1)A$ have nonzero global sections, and choose $F\in |qA|$. Condition (1) and the section subtraction lemma for big divisors yield an integer $j>0$ with $jD-qA\sim E_0\ge 0$. Adding an effective member of $|(q-1)A|$ gives $jD\sim A+E$. Thus (1) implies the stronger condition (2').

If $jD\sim A+E$ with $A$ ample and $E$ effective, multiplication by the section of $qE$ injects $H^0(X,qA)$ into $H^0(X,qjD)$. The positive leading coefficient of the ample Hilbert polynomial supplies condition (1) along the infinite sequence $m=qj$. Condition (2) implies (3). Conversely, if $jD\equiv A+E$, put $P=jD-E$. The Cartier divisor $P$ is numerically equivalent to $A$, hence ample by Kleiman's criterion; the actual equality $jD=P+E$ gives (2).

We have proved (1)$\Rightarrow$(2')$\Rightarrow$(2)$\Rightarrow$(1), and (3)$\Rightarrow$(2)$\Rightarrow$(3). Also (2') implies (3'), while (3') implies (3). Hence all five conditions are equivalent. They characterize a big divisor; the ample-plus-effective formulation is Kodaira's lemma. The inconsistent use of $n$ and $j$ in the printed multiplier is resolved by using $j$ throughout.

For an effective Cartier divisor $F$, the divisor restriction exact sequence givesThe 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 $C{k}^{n-1}$. For the infinitely many $k$ in the hypothesis,once $k$ is sufficiently large. Thus 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.