My question arises from the proof of theorem 1.4 in https://cpb-us-w2.wpmucdn.com/sites.bc.edu/dist/a/54/files/2019/12/nefcone.pdf
For a projective klt pair $(X,\Delta)$, the Mori cone theorem says that $$\overline{NE}(X)= \overline{NE}(X)_{K_X+\Delta \ge 0} + \sum \mathbb{R}_{\ge 0} [C_j] $$ For countably many rational curves $C_j$. In the first paragraph of the proof, we have a divisor $D$ such that $D$ is positive on $\overline{NE}(X)_{K_X + \Delta\ge 0}$, i.e $D\cdot C > 0$ for all $[C]\in \overline{NE}(X)_{K_X+\Delta \ge 0}$, and negative somewhere on $\overline{NE}(X)$.
In what follows, the author claims that after rescaling, $D = K_X + \Delta +A$ for some ample divisor. My question is, how do we arrive at the existence of such an ample $A$? I can only show that $D - a(K_X + \Delta)$ is a divisor positive on $\overline{NE}(X)_{K_X+\Delta \ge 0}$ for some sufficiently small $a>0$. To show that it is ample, I have to show that it is positive on the whole cone $\overline{NE}(X)$. How do I do this?
Any hints/help given should be greatly appreciated! Let me know if more details are needed.