Exercise 16, § 1

Let E be a vector space of finite dimension, and $f$ a bounded function on $I \times \mathbf{H}$ satisfying the conditions of IV, p. 165, lemma 1, such that the equation $x' = f(t, x)$ has a unique solution $u$ defined on $I$, with values in $\mathbf{H}$, equal to $x_0$ at the point $t_0$. Suppose further that for sufficiently large $n$ there is an approximate integral $u_n$ of $x' = f(t, x)$ to within $1/n$, defined on $I$, with values in $\mathbf{H}$, equal to $x_0$ at the point $t_0$. Show that the sequence $(u_n)$ converges uniformly to $u$ on every compact interval contained in $I$ (use the fact that the sequence $(u_n)$ is equicontinuous on $I$).