Let $\gamma: (a,b) \rightarrow \mathbb{R}^d$ be the maximal solution of the differential equation $x' = F(t,x)$ with initial condition $\gamma(t_0)=x_0$. Show that given any compact $K \subset I$ and any $\epsilon > 0$, there exists $\delta > 0$ such that for every $(\overline{t}, \overline{x}) \in B_{\delta} ((t_0, x_0))$ the domain of the maximal solution $\beta$ with initial condition $\beta(\overline{t}) = \overline{x}$ contains the compact $K$ and $||\beta(t) - \gamma(t)|| \leq \epsilon$, for $t \in K$.
Attempt: I considered an open cover $\mathcal{V}$ of the set $K\times B_{\epsilon}(x_0)$, given by $\mathcal{V} = \{B_{\rho}(t_0)\times B_{\rho}(x_0); (t_0, x_0) \in K\times\overline{B_{\epsilon}(x_0)}\}$, with $\rho > 0$ given by the Continuous Dependence Theorem (assuming that $F$ is continuous and locally lipschitz). Let $\delta > 0$ be the Lebesgue number of the cover $\mathcal{V}$. Therefore, for every $(\overline{t}, \overline{x}) \in K\times \overline{B_{\epsilon}(x_0)}$, there exists a $t_0$ such that $(t_0, x_0) \in K\times\overline{B_{\epsilon}(x_0)}$ and $B_{\delta}(\overline{t})\times B_{\delta}(\overline{x}) \subset B_{\rho}(t_0) \times B_{\rho}(x_0)$. From this, we can conclude that $\left[\overline{t} - \delta , \overline{t} + \delta\right] \subset Dom({\beta_{\overline{t}, \overline{x}}})$, with $\beta_{\overline{t}, \overline{x}}$ the maximal solution of $x' = F(t,x)$ with initial condition $\beta_{\overline{t}, \overline{x}}(\overline{t})= \overline{x}$. How do I proceed?
Any help would be appreciated!