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Suppose that $f : [a, b] \to \mathbb{R}$ is continuous and that $f([a, b]) \subset [a, b]$. Prove that there exists a point $c \in [a, b]$ satisfying $f(c) = c$.

(If either $f(a) = a$ or $f(b) = b$ there is nothing left to show, so you might as well assume that $f(a) = a$ and $f(b) = b$. Since $f$ takes its values in $[a, b]$ this is the same as assuming that $f(a) > a$ and $f(b) < b$.)

So far, I have:

Pf. Assume $f(a)>a$ and $f(b)< b$. Let $x, y \in [a,b]$ such that $f(a)=x$ and $f(b)=y$ which means $f[a,b]=[x,y]$. Notice $[x,y]\subset [a, b]$. Since f is continuous on $[x,y]$, there exists some $c \in [a,b]$ such that $x$ is less than or equal to $c$ is less than or equal to $y$...

This is where I am stuck because I don't think I can just assume by Intermediate Value Theorem that some $f(c)=c$?

Answers