TITLE: Any homeomorphism from $[0,1)\to [0,1)$ has a fixed point.
QUESTION [6 upvotes]: Show that 

Any homeomorphism from $[0,1)\to [0,1)$ has a fixed point.

My try:
Suppose that  $f(x)\neq x$ for all $x$,then either $f(x)>x$ or $f(x)<x$ for all $x$ otherwise if $f(a)>a$ and  $f(b)<b$ for some  $a,b\in [0,1)$ then by IVP $f(p)=p;p\in [a,b]\subset [0,1)$ which is false.
Hence take WLOG ; $f(x)>x$ for all $x$ ;
Also if $f^{-1}(x)>x$ then by above we have $f(f^{-1}(x))>f^{-1}(x)>x\implies x>x $ false .Hence $f^{-1}(x)\le x$ for some $x$ .
But I can't complete the proof from here.Please give some hints so that I can take it forward.

REPLY [2 votes]: Let $I_1$ and $I_2$ be intervals on the real line.  Let $f$ be a homeomorphism between them.

Use IVP (the Intermediate Value Property) to show that $f$ is either increasing or decreasing.

Deduce that $f$ takes endpoints of $I_1$ to endpoints of $I_2$.

Conclude that if $I_1 = I_2 = [0,1)$, then...