Prove that if function f is monotonic, then it one-to-one What I have so far:
Suppose $f$ is monotonic. It is therefore either increasing or decreasing.
Proof for increasing:
If $f$ is increasing, then $f(x_1) <f(x_2)$ whenever $x_1 < x_2$, which means $f(x_1) = f(x_2)$ if, and only if, $x_1 = x_2$.
Therefore $f$ is one to one.
I think this is wrong though. Could anyone help me out?

Here's a nice way to phrase the argument for $f$ increasing.
$$\begin{align}
f(x)=f(y) & \iff f(x)\le f(y)\text{ and }f(x)\ge f(y)\\
& \iff x\le y\text{ and }x\ge y\\
& \iff x=y.
\end{align}$$
Come to think of it. This exact argument works for $f$ decreasing.
