How to prove the 1 Lipschitz function defined on the closed unit ball has a fixed point

Question

I need to prove that the 1 Lipschitz function has a fixed point:

$\|f(x)-f(y)\|≤ \|x-y\|$ for all $x,y\in B$, where $B$ is the closed unit ball in the $R^n$.

I want to apply the contraction mapping theorem. The contraction mapping theorem is not suitable for 1 Lipschitz function because it requires that $\lambda \in [0,1)$, so I want to modify this function to obtain a contraction. Anyone could please gives me some hints about how to get such contraction?

Answer 1

Note that $(1-{1 \over n}) f$ is $(1-{1 \over n})$ Lipschitz and so has a fixed point $x_n$ that satisfies $(1-{1 \over n}) f(x_n) = x_n$. Since $B$ is compact, $x_n$ has an accumulation point $x^*$ and continuity of $f$ gives $f(x^*) = x^*$.