  We define the infinite dimensional simplex to be the closure of the convex
hull of the standard basis vectors in R^infinity, and prove that this space has
the 'fixed point property': any continuous function from the space into itself
has a fixed point. Our proof is constructive, in the sense that it can be used
to find an approximate fixed point; the proof relies on elementary analysis and
Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed
point theorem on infinite-dimensional compact convex subsets of normed spaces.
