We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$. Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$.
Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that:
$$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$