Definition 4

Tag 08E6, permanent. Cite it as [Bourbaki, Tag 08E6].

Functions of a Real Variable, chapter V, § 1. COMPARISON OF FUNCTIONS ON A FILTERED SET

Given two functions $f, g$ of $\mathscr{H}(\mathfrak{F}, V)$ one says that $f$ and $g$ are equivalent (along $\mathfrak{F}$ ), and writes $f \sim g$ , if $f - g \ll f$ .

The relation $f \sim g$ implies $f \asymp g$ but is not equivalent to it.

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

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