TITLE: Fixed point of a monotone on [0,1].
QUESTION [6 upvotes]: prove or disprove: Let $f:[0,1]\rightarrow[0,1]$ be a monotone (need not be strict) function then  f has a fixed point. 
Can I have a hint?

REPLY [4 votes]: This might be a somewhat simpler argument.
Suppose that $f:[0,1]\mapsto[0,1]$ is non-decreasing and assume that $f$ has no fixed point.
Define
$$
s=\sup\{x:f(x)\gt x\}
$$
Note that $0\in\{x:f(x)\gt x\}$, so $s\in[0,1]$.
If $f(s)\gt s$, then for all $x\in(s,f(s))$, we must have $\ f(x)\lt x$ by the definition of $s$. But then $f(x)\lt x\lt f(s)$ contradicts that $f$ is non-decreasing.
If $f(s)\lt s$, then, by the definition of $s$, there must be some $x\in(f(s),s)$ so that $f(x)\gt x$. But then $f(s)\lt x\lt f(x)$  contradicts that $f$ is non-decreasing.
Thus, the assumption that $f$ has no fixed point must be false.