Let $M$ be a riemanninan manifold of dimension $n$ with connection $D$. Given $p \in M$, I want to prove that there're a neighborhood $V$ of $p$ and $\varepsilon, \delta>0$ such that for every $q \in V$ and for every $X \in T_qM$ with $||X||<\varepsilon$ there exists a geodesic $\gamma\colon (-\delta,+\delta) \to M$ with $\gamma(0)=q$ and $\dot{\gamma}(0)=X$.
Let $(U\varphi)$ be a chart near $p$. Consider the system of ODE
$\cases{\dot{u_k}=v_k \\ \dot{v_k}=-\sum_{i,j=1}^n \Gamma_{ij}^k v_i v_j}$
for $k=1,...,n$. This is of the form $\dot{w}=F(w)$ with $F\colon \Omega \to \mathbb R^{2n}$ where $\Omega=\varphi(U) \times \mathbb R^n$.
From the theory of ODEs I know that there are an open neighborhood $\Omega_0$ of $(\varphi(p),0)$ contained in $\Omega$ and a number $\delta>0$ such that the Cauchy problem given by the system of ODE and the initial condition $w(0)=(x_0,y_0) \in \Omega_0$ (arbitrarly chosen) admits a unique solution $w\colon (-\delta, \delta) \to \mathbb R^{2n}$.
Question: for sure there're $W$ and $A$ open neighborhood of $\varphi(p)$ and $0$ respectively such that $W \times A \subseteq \Omega_0$, so that I can put $V=\varphi^{-1}(W)$, but how can I find $\varepsilon>0$ such that for every $q \in V$ and for every $X \in T_qM$ with $||X||<\varepsilon$ it holds that $\mathrm d_q \varphi (X) \in A$? Should I use continuity of the map $X \mapsto <X,X>$?