Brouwer Fixed Point Theorem: A Proof for Economics Students

2 Proof
Let f(x) be a map from a compact convex set X into itself. We
assume that f(x) is twice continuously differentiable and that X
is a set in the n-dimensional Euclidean space R n (n ≥ 1) defined
by
X ≡ {x| x ≥ 0, X n
x i ≤ 1, x∈ R n }.
i=1
Then defineadirectproductset Y
Y ≡ X × T, where T ≡ [0, ∞) .
The symbol J f (x) denotes the Jacobian matrix of the map f(x).
Now we begin our proof. First of all, as the hypothesis of mathematical
induction, we suppose that the theorem is true when the
dimension is less than n. (Whenn =1, it is easy enough to show
the existence of at least one fixed point.) Let us consider the set
C ≡ {y| f(x) − t · x =0,y=(x, t) ∈ Y }.
2