# How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant.

$$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$

where $f = \cos \circ \tan \circ \sin$

How do I formally prove this?

• It is enough to show that there is a constant $c<1$ such that for all $x$, $|f'(x)|<c$, because then the Mean Value Theorem implies that $f$ is a contraction mapping. – Brent Kerby Feb 24 '15 at 0:46