Remark

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

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

Note that the relation $f \sim g$ by no means implies that the difference $f - g$ tends to 0 along $\mathfrak{F}$ ; this difference can even be unbounded, as is shown by the example $x^2 + x \sim x^2$ as $x$ tends to $+\infty$ .

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

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