Unit-Speed Minimizing Curve is Geodesic

Theorem
Let $\struct {M, g}$ be a Riemannian manifold.

Let $I = \closedint a b$ be a closed real interval.

Let $\gamma : I \to M$ be an admissible curve.

Suppose $\gamma$ is parametrized so that $\gamma$ is a unit-speed curve.

Suppose $\gamma$ is minimizing.

Then $\gamma$ is a geodesic.