A compact-target boundary point has a null geodesic segment

Apply [F4] in $(E,g)$ to the pair provided by [F3]. Global hyperbolicity holds, and the endpoints are distinct and causally related. In the source’s section 2.1 convention this gives a nonconstant future causal affine geodesic segment $\gamma :[a,b]\to E$, $a<b$, with $\gamma (a)=p$ and $\gamma (b)=c$; both endpoints are included in the segment.

By [F6] and the affine geodesic equation, $\frac{d}{ds}g(\dot{\gamma },\dot{\gamma })=2g(D_s\dot{\gamma },\dot{\gamma })=0$. Thus the squared speed is constant. If it were negative, the future causal geodesic would be timelike from $p$ to $c$, contrary to [F3]. Its tangent is therefore null everywhere, with the nonzero future tangent required in [F5].

Put $\eta (t)=\gamma (a+(b-a)t)$ for $0\le t\le 1$. The chain rule gives $\eta \mathrm{\prime }=(b-a)\dot{\gamma }$ and $D_t\eta \mathrm{\prime }=(b-a{)}^2{D}_s\dot{\gamma }=0$, so this is the desired positive affine normalization. The imported segment has included endpoints, rather than merely an asymptotic endpoint. In endpoint charts its equation is $x^{\mu \mathrm{\prime }\mathrm{\prime }}=-{\mathrm{\Gamma }}_{\alpha \beta }^{\mu }(x)x^{\alpha \mathrm{\prime }}{x}^{\beta \mathrm{\prime }}$, with smooth coefficients by [F6]; starting with its twice differentiable geodesic solution, this equation successively gives derivatives of every order, with one-sided derivatives at included endpoints. Hence $\gamma$ and its affine reparametrization are smooth up to both endpoints, as asserted.