Example 1

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

General Topology, chapter IV, § 7. Infinite sums and products of real numbers

Let $a$ be a finite real number $> 0$ , and consider the sequence $\left( \frac{a^n}{n!} \right)$ ; let $n_0$ be the smallest integer such that $a < n_0$ . Then for each $n \geq n_0$ we have

\frac{a^n}{n!} \leq \frac{a^{n_0}}{n_0!} \cdot \left( \frac{a}{n_0} \right)^{n-n_0};

since

q = \frac{a}{n_0} < 1

, the sequence

(q^{n-n_0})

is summable, and therefore so is the sequence

\left( \frac{a^n}{n!} \right)

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

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