I just want to add that I also assume $\mathbb{R}^n$ with respect to the metric $G$ is geodesically complete. Hence, for computing $\mathbf{d}_G$, I am just taking integral along the geodesic.

By having that lower bound, I can show that any Cauchy sequence in $\mathbb{R}^n$ with respect to $d_G$ is a Cauchy sequence with respect to $d_g$, where $g$ is Euclidean metric or the flat metric. As we know, $\mathbb{R}^n$ with respect to $d_g$ is a complete metric space. Hence, the sequence converge to a point in $\mathbb{R}^n$ with respect to $d_g$. Now, how can I show that the sequence converges to the same point with respect to $d_G$?

How can I show the result using this condition? Do I need to construct a Cauchy sequence on the original metric and show that it has a limit?