Lemma 10. Every continuous self-map g of a nonempty compact convex subset K of a normed space has a fixed point.

For δ>0 , choose a finite strict δ -net P_1,…,P_N∈K . For x∈K set λ_i(x)= δ-P_i-g(x),0 ∑_jδ-P_j-g(x),0 . The denominator is positive, and these are continuous probability weights. Map the probability simplex to itself by sending coefficients (a_i) to (λ_i(∑_j a_jP_j)) . By Lemma 9 there is a fixed coefficient vector. Its convex combination x∈K satisfies x-g(x) ≤∑_iλ_i(x)P_i-g(x)<δ, since a positive weight implies the corresponding distance is less than δ . Letting δ0 and taking a convergent subsequence in K gives a fixed point.