Let $M$ be a smooth manifold with connection, and fix a point $p$. We know that geodesics exist locally in the following sense; given any $X \in T_p M$, there is an open interval $I$ containing zero, and a geodesic $\gamma_X: I \to M$ such that $\gamma_X(0) = p$ and $\dot{\gamma}_X(0) = X$.
How do we show that there is an open set $U$ arond $p$ such that for all $y \in U$, there is a geodesic between $p$ and $y$?
It seems this is implicit in the construction of the exponential map, thus normal coordinates. I've tried thinking in terms of basic topological considerations and intuition, but can't turn this into a rigorous proof. Is this result a consequence of the existence result above?