Corollary

Tag 03RK, permanent. Cite it as [Bourbaki, Tag 03RK].

Theory of Sets, chapter III, § 5. Properties of integers

For each integer $n > 0$ , we have

\sum_{i=1}^{n} i = \frac{1}{2} n(n+1).

In the set A of ordered pairs of integers $(i, j)$ such that $1 \leqslant i \leqslant j \leqslant n$ , let $A_k$ denote the subset of pairs $(i, k)$ , where $1 \leqslant i \leqslant k$ (for an arbitrary integer $k \leqslant n$ ). Then $A_k$ has $k$ elements. But $(A_k)_{1 \leqslant k \leqslant n}$ is a partition of A; hence the result.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Exercise 1, § 5

Cites

Nothing.