I want to understand the proposition 3.7 from the note here:
Proposition 3.7. Let $(M, g)$ be a globally hyperbolic spacetime, S a Cauchy hypersurface with future-pointing unit normal vector field $n$, $\Sigma \subset S $ a compact 2-dimensional submanifold with unit normal vector field $\nu$ in $S$, $p \in \Sigma$, $c_p$ the null geodesic through $p$ with initial condition $n_p + ν_p$ and $q = c_p(r)$ for some $r > 0$. If $c_p$ has a conjugate point between $p$ and $q$ then $q ∈ I^+(\Sigma)$.
Simply, a conjugate point is where two geodesics intersect. I don't understand the proof because the piecewise curve constructed by gluing $\gamma$ and $c_p$ together is not time-like, it is everywhere null but it is claimed it is in $I^{+}(\Sigma)$.