Brouwer's fixed point theorem says that any continuous function $f$ mapping a compact convex set $\mathbb{\Omega}$ to itself has a fixed point. The other day I was reading a piece in a pop-sci magazine, which asked to prove the theorem in $\mathbb{R}$. I got this: could you please confirm if it's correct, and in case it's not, help me find a valid proof? Also, if you have a more elegant proof, I'd love to hear about it. Thanks!
A compact convex set in $\mathbb{R}$ is a closed interval, so I need to prove this for $\mathbb{\Omega}=[a,b]$. Now, let's consider $g(x)=f(x)-x$, which is also continuous. We assume $g(a)\cdot g(b)\neq0$, otherwise we already found a fixed point. Of course, we must have $g(a)>0$, otherwise we would have $f(a)<a$ which contradicts the hypothesis that $f$ maps $[a,b]$ into itself. Similarly, we must have $g(b)<0$. Since $g(a)\cdot g(b)<0$, then by Bolzano's theorem there is at least one $x_0\in[a,b]$ such that $g(x_0)=0\implies f(x_0)=x_0\ \square$
Best Answer
Yes, your proof is correct.
It is a standard proof. It is the one in the wikipedia article.