Regular Curve in Riemannian Manifold has Unit-Speed Forward Reparametrization

Theorem
Let $\struct {M, g}$ be a Riemannian manifold with or without boundary.

Let $I \subseteq \R$ be a real interval.

Let $\gamma : I \to M$ be a regular curve.

Then $\gamma$ has a unit-speed forward reparametrization.