Example

Tag 0C4K, permanent. Cite it as [Bourbaki, Tag 0C4K].

In the space $\mathbf{R}^n$ , the three norms

\sqrt{\sum_{i=1}^n x_i^2}, \quad \sup_{1 \leq i \leq n} |x_i| \quad \text{and} \quad \sum_{i=1}^n |x_i|

are equivalent, because we have (2)

\sup_{1 \leq i \leq n} |x_i| \leq \sqrt{\sum_{i=1}^n x_i^2} \leq \sum_{i=1}^n |x_i| \leq n . \sup_{1 \leq i \leq n} |x_i|.

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

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