Exercise 1, § 1

a) With the notation of IV, p. 165, suppose that $f$ is uniformly continuous on $J \times S$. Prove prop. 3 of IV, p. 166 without using the axiom of choice. (Let $\delta$ be such that the relations $|t_2 - t_1| \leq \delta, \|x_2 - x_1\| \leq \delta$ imply $\|f(t_2, x_2) - f(t_1, x_1)\| \leq \varepsilon$; consider a subdivision of $J$ into intervals of length $\leq \inf(\delta, \delta/M)$ and define the approximate solution by induction.)

b) When $E$ is finite dimensional and $f$ is Lipschitz on $I \times H$ prove prop. 3 of IV, p. 166 without using the axiom of choice. (Remark that for every $\delta > 0$ there exists a finite number of points $x_i$ of $S$ such that every point of $S$ is at a distance $\leq \delta$ from one of the $x_i$; then proceed as in $a$), considering the regulated functions $t \mapsto f(t, x_i)$, of which there are only a finite number.)