Suppose $S$ is an achronal set in a spacetime $M$. And $S$ is closed. At the same time, any null geodesic of $M$ intersects $S$. Then, why does any timelike curve from $I^+[S]$ to $I^-[S]$ intersect $S$, too?
I understand that, any point $r\in M$ belongs to either $S$, or $I^+[S]$ or $I^-[S]$. Because if $r\notin S$, and there is a past null geodesic $\gamma$ starting at $r$, then $\gamma$ must intersect $S$ at $q$. Pick any point $p\in\gamma$ and $p$ is in the causal future of $q$, then there is a second null geodesic $\eta$ from $p$ and intersecting $S$ at $s$. Therefore, we can find a timelike curve connecting $r$ to $s$ which implies that $r\in I^+[S]$.
But unfortunately, I cannot figure out how to show any timelike curve from $I^+[S]$ to $I^-[S]$ intersects $S$. Would you give me some hint, please?