An Intuitive Proof of Brouwer's Fixed Point Theorem in R2

Abstract

Fixed point theorems play a major role in general equilibrium theory. Brouwer's theorem is the most basic of these; it states that any continuous function mapping a closed bounded convex set into itself must contain at least one fixed point (i.e., a point that is its own image). Elementary discussions invariably give an intuitive proof of the theorem for functions of a single variable, as illustrated in FIGURE 1. In Rl a set is convex if and only if it is an interval; thus a continuous mapping of the closed bounded interval [x0, xl] into itself can be represented by a curvef. Sincef connects the left-hand side of the rectangle to the right-hand side of the rectangle, it is intuitively obvious that f must intersect the diagonal of the rectangle at least once, and at this pointf(x*)=x*. A bit more formally, iff(xo) * xo andf(xl) *xl, then O(xo) =f(xo) xo > 0 and 4)(x1) =f(x ) xl < 0. Since 4 is continuous on [xo, xl], the intermediate value theorem implies that 4 must assume the value zero somewhere on the open interval (xo, xl), which proves the theorem. An intermediateor advanced-level student should be a bit street-wise and skeptical of the validity of demonstrations based on two-dimensional diagrams. The purpose of this note is to demonstrate that the intuitive graphic proof generalizes to three dimensions (i.e., to functions on R2) and can be made rigorous at that level. To begin, let W be any closed bounded (i.e., compact) convex set in R2 and let f be any continuous function mapping W into itself. Since W is bounded it can be contained in a rectangle as shown in FIGURE 2. We may now extendf to the closed rectangle ABCD as follows. Choose an arbitrary interior point a in W and for each point b in the rectangle but not in W, define f(b) to be the image of the point c at which the line through a and b intersects the boundary of W. The