. Assume that there exist a pseudoeffective $\Q$-divisor $F$ on $X$ and an infinite subset $\mathcal S\subseteq\N$ such that $$\label{eq:rel0a} N_m+F\sim_\Q m(K_X+\Delta)$$ for all $m\in\mathcal S$, where $N_m\geq0$ are integral Weil divisors. Then $$\kappa(X,K_X+\Delta)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq0.$$  *Step 0.* Note first that implies $$\label{eq:rel0b} N_p-N_q\sim_\Q (p-q)(K_X+\Delta)\quad\text{for all }p,q\in\mathcal S