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

Let $I_1$ and $I_2$ be intervals on the real line. Let $f$ be a homeomorphism between them.

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

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

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