Remark

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

Functions of a Real Variable, chapter V, Appendix 0. Hardy Fields. (H) Functions

This very elementary property does not extend to linear differential equations of order $> 1$ ; for example, the function $y = \sin x$ satisfies $y'' + y = 0$ , but changes sign on every neighbourhood of $+\infty$ .

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

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