Definition 1

Tag 0A4H, permanent. Cite it as [Bourbaki, Tag 0A4H].

Lie Groups and Lie Algebras, chapter II, § 4. Central filtrations

Let $G$ be a group. A real filtration on $G$ is a family $(G_\alpha)_{\alpha \in \mathbf{R}}$ of subgroups of $G$ such that

G_\alpha = \bigcap_{\beta < \alpha} G_\beta \quad \text{for all } \alpha \in \mathbf{R}.

Formula (1) implies $G_\alpha \subset G_\beta$ for $\beta < \alpha$ and hence the family $(G_\alpha)$ is decreasing. The filtration $(G_\alpha)$ is called separated if $\bigcap_\alpha G_\alpha$ reduces to the identity element and is called exhaustive if $G = \bigcup_\alpha G_\alpha$ .

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

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