I am studying the global causality of the spacetime. Here, I come across a problem.
Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in spacetime. Then, it is claimed that $I^+(r)\subset I^+(p)$. But why?
Let me first try to prove this conclusion. I notice there is a theorem:
Let a subset $S\subset M$ ($M$ the spacetime manifold) and set $B=\partial I^+[S]$. Then, if $x\in B-\bar S$, there exists a null geodesic $\eta\subset B$ with future endpoint $x$ and which is either past-endless or has a past endpoint on $\bar S$.
So we can set $S=\{p\}=\bar S$. Since $r\in\partial I^+(p)$ and $r\ne p$, $r\in B-\{p\}$ with $B=\partial I^+(p)$. Therefore, there is a null geodesics $\eta$ lying on $B$ and passing through $p$. Is this correct?