Radial Geodesic Connecting Two Points in Geodesic Ball is Unique Minimizing Curve

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

Let $U = \map {\exp_p} {\map {B_\epsilon} 0 }$ be a geodesic ball centered at $p \in M$.

Suppose $q \in U$.

Suppose $\gamma$ is a radial geodesic from $p$ to $q$.

Then, up to reparametrization, $\gamma$ is the unique minimizing curve from $p$ to $q$.