Proposition 6

*Let $\mathbf{f}$ and $\mathbf{g}$ be two functions defined on $I \times H$, satisfying the hypotheses of lemma 1 of IV, p. 165, and such that, on $I \times H$,

(16)

Suppose further that $\mathbf{g}$ is Lipschitz for the constant $k > 0$ on $I \times H$. In these circumstances, if $\mathbf{u}$ is an approximate solution of $\mathbf{x}' = \mathbf{f}(t, \mathbf{x})$ to within $\varepsilon_1$, defined on $I$, with values in $H$, and $\mathbf{v}$ is an approximate solution of $\mathbf{x}' = \mathbf{g}(t, \mathbf{x})$ to within $\varepsilon_2$, defined on $I$, with values in $H$, then, for all $t \in I$

(17)

Indeed

for all $t$ in the complement in $I$ of a countable subset of $I$; in other words, $\mathbf{u}$ is an approximate solution of $\mathbf{x}' = \mathbf{g}(t, \mathbf{x})$ to within $\alpha + \varepsilon_1$, so the inequality (17) follows on applying prop. 5 of IV, p. 169.