 Solution

Suppose $D\equiv A^{\prime }+E^{\prime }$ as in (c), and set $P=D-E^{\prime }$. Its numerical class is ample, so $P$ is an ample real divisor. Fix an ample Cartier divisor $A$. The ample cone is open by Kleiman's criterion, so choose $\epsilon >0$ with $P-\epsilon A$ still ample. Express this ample real divisor as a positive real combination $\sum b_i{B}_i$ of ample Cartier divisors. Some positive multiple of each $B_i$ has an effective representative; dividing by that multiple gives $B_i{\sim }_{\mathbb{Q}}{G}_i$ with $G_i$ effective. Hencewhich is (b). The equivalence of numerical ampleness with a positive combination of ample Cartier representatives follows by rational approximation inside the open ample cone; the rational approximation of an ample-plus-effective real divisor proof also accounts for principal-divisor directions.

Finally, if $D^{\prime }\equiv D$, the same expression $D^{\prime }\equiv A^{\prime }+E^{\prime }$ satisfies (c). By the equivalences just proved, $D^{\prime }$ satisfies all three conditions exactly when $D$ does. Thus bigness is invariant under numerical equivalence, for real as well as Cartier divisors.