Proposition 4

Tag 08BI, permanent. Cite it as [Bourbaki, Tag 08BI].

Functions of a Real Variable, chapter IV, § 1. EXISTENCE THEOREMS

The set of approximate solutions of (1) to within $\varepsilon$ , defined on the same interval $K \subset J$ and taking values in S, is uniformly equicontinuous.

Indeed, if u is any function in this set and s and t are two points of K, then by the mean value theorem

\|u(s) - u(t)\| \leq (M + \varepsilon)|s - t|.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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