Geodesic Ball as Metric Ball in Riemannian Manifold

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

Let $U^c$ and $U^o$ be closed and open geodesic balls with radii $R_c$ and $R_o$ respectively in $M$.

Then $U^c$ is a closed metric ball with radius $\epsilon_c = R_c$, and $U^o$ is an open metric ball with radius $\epsilon_o = R_o$.