PLEASE DO NOT COPY PASTE FROM CHEGG/OTHERWISE I'LL REPORT YOU OR GIVE OU A DISLIKE

Consider the sequence of functions as defined in the proof of the Existence and Uniqueness Theorem, i.e., $\mathrm{X}_{0}(t)=\mathrm{x}_{0}$ $$ \mathbf {x}_{n+1}(t)=\mathbf {x}_{0}+\int_{0}^{t} \mathbf {f}\left(\mathbf {x}_{n}(5) ight) ds, n=0,1, \ldots $$ Show that if $\mathbf {x}_{n}(t)$ is defined, continuous and belongs to the compact set $Dfor all $t \in[-\alpha, \alpha]$ and $\mathbf {f} \in C^{1} D$, then $\mathbf{x}_{n+1}() is defined and continuous for all $t \in[-\alpha, \alpha]$.CS.SD. 179