Let $(M, g)$ be a globally hyperbolic spacetime with $\Sigma$ a Cauchy surface. It is well known that such spacetimes can be geodesically incomplete. But the following statement seems intuitive and, if it is true, seems like it might appear in an introduction to causality theory/globally hyperbolic spacetimes, though I cannot find it:
Let $p \in M$ lie to the future of $\Sigma$. Let $V$ be a past-directed causal vector in $T_p M$. Let $\gamma$ be the unique maximal geodesic starting at $p$ with initial velocity $V$. Then $\gamma$ intersects $\Sigma$.
Since $\Sigma$ is a Cauchy hypersurface, any past-directed causal curve emanating from such a point $p$ which is inextendible as a causal curve must intersect $\Sigma$. The $\gamma$ we have here is inextendible as a geodesic, but I am not sure if it is inextendible as a causal curve.