Prove that the rotation number exists and is well-defined

This is commonly taught in the first semester of a dynamical systems course. Say we have a continuous, increasing function $F : \mathbb{R}\to \mathbb{R}$ with the property that $F(x+1) = F(x)+1$ for all $x\in \mathbb{R}$. Prove that \[\omega(F) = \lim_{n\to \infty} \frac{F^n(x)}{n}\] exists and is independent of $x$. Here, $F^n(x) = (F\circ F\circ\cdots \circ F)(x)$, with $n$ copies of $F$ on the right-hand side.

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$