I was reading this question [Limit of $(\sin\circ\sin\circ\cdots\circ\sin)(x)$](https://math.stackexchange.com/questions/58677) and nowhere did it talk about how to prove the existence of the limit.

In general, if you're given a function $f(x)$, how do you check for the existence of $\lim\limits\_{n\to \infty} \underbrace{(f\circ f \circ \cdots \circ f)}\_{n\text{ times}}(x)$?

When $f(x) = \sin(x)$, if the limit $c$ exists , then $f(c)$ must equal $c$ so $c = 0$ right? But isn't it also possible that the limit doesn't exist?

Or is it the case that when you have a monotonically decreasing function that's bounded below by some value, then $\lim\limits\_{n\to\infty} \underbrace{(f \circ f \circ \cdots \circ f)}\_{n\text{ times}}(x)$ is defined and necessarily a value that satisfies the equation $f(x) = x$?

Oh, and as a side note, is there a better way of denoting $f(x)$ composed with itself $n$ times?