Remark 1

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

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

When $k = 0$ the inequality (15) is replaced by the inequality

\| \mathbf{u}(t) - \mathbf{v}(t) \| \leq \| \mathbf{u}(t_0) - \mathbf{v}(t_0) \| + (\varepsilon_1 + \varepsilon_2) |t - t_0|

whose proof is immediate.

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

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