### 13.6 Geodesic completeness

The analysis described above concerned only the functions and their averages. In particular, no
estimates for the derivatives in the sup norm were derived. However, in order to prove future causal
geodesic completeness, it is of interest to have such estimates. According to [75, Proposition 1.8,
p. 665]:
Proposition 6 Consider a solution to Equations (11)–(12). Then

Using this estimate, it is then possible to prove future causal geodesic completeness; see [75, Theorem 1.9,
p. 665].