Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$.
Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence: \begin{equation} \{p_i := f(G_{f^{-1}[p_{i-1}]}^{-1}(V)+ f^{-1}[p_{i-1}]) \}_{n \in \mathbb{N}}, \end{equation} where $p_0=p$ and $G_{f^{-1}[p_{i-1}]}^{-1}$ is the inverse of the pull-back of the metric tensor at $f(p)$ to $p$ in $U$.
Then it seems intuitive to me that $\cup_{n \in \mathbb{N}}\{f(p_i)\}$ converges to the image of geodesic in $M$ with velocity $V$, but how can I prove this?
I was thinking of taking a discrete derivatives but still this is not obvious to me. Thank you.