Editing Brouwer fixed-point theorem (section)

===A combinatorial proof===
The BFPT can be proved using [[Sperner's lemma]]. We now give an outline of the proof for the special case in which ''f'' is a function from the standard ''n''-[[simplex]], <math>\Delta^n,</math> to itself, where

:<math>\Delta^n = \left\{P\in\mathbb{R}^{n+1}\mid\sum_{i = 0}^{n}{P_i} = 1 \text{ and } P_i \ge 0 \text{ for all } i\right\}.</math>

For every point <math>P\in \Delta^n,</math> also <math>f(P)\in \Delta^n.</math> Hence the sum of their coordinates is equal:

:<math>\sum_{i = 0}^{n}{P_i} = 1 = \sum_{i = 0}^{n}{f(P)_i}</math>

Hence, by the pigeonhole principle, for every <math>P\in \Delta^n,</math> there must be an index <math>j \in \{0, \ldots, n\}</math> such that the <math>j</math>th coordinate of <math>P</math> is greater than or equal to the <math>j</math>th coordinate of its image under ''f'':

:<math>P_j \geq f(P)_j.</math>

Moreover, if <math>P</math> lies on a ''k''-dimensional sub-face of <math>\Delta^n,</math> then by the same argument, the index <math>j</math> can be selected from among the {{nowrap|''k'' + 1}} coordinates which are not zero on this sub-face.

We now use this fact to construct a Sperner coloring.  For every triangulation of <math>\Delta^n,</math> the color of every vertex <math>P</math> is an index <math>j</math> such that <math>f(P)_j \leq P_j.</math>

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an ''n''-dimensional simplex whose vertices are colored with the entire set of {{nowrap|''n'' + 1}} available colors.

Because ''f'' is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point <math>P</math> which satisfies the labeling condition in all coordinates: <math>f(P)_j \leq P_j</math> for all <math>j.</math>

Because the sum of the coordinates of <math>P</math> and <math>f(P)</math> must be equal, all these inequalities must actually be equalities. But this means that:

:<math>f(P) = P.</math>

That is, <math>P</math> is a fixed point of <math>f.</math>