Limit of infinite composition of sin(x) I was playing around on desmos the other day, and noticed that $\sin\left(\sin\left(x\right)\right)$ is basically a version of sin with a lower amplitude (which makes intuitive sense). To me, it seems intuitve that this curve, when composed infinite times, becomes a straight line, as the values at $x=\frac{\pi}{2}\mathbb{Z}$ would move (slowly) towards the values at $x=\pi\mathbb{Z}$ by virtue of them moving away from the peaks, but is there a way to go about properly proving this?

We know that $\sin(\mathbb{R})=[-1,1]$  Since the sine function is odd, it will be enough to consider the interval $[0,1]$.  The sine function is increasing on this interval so $\sin([0,1])=[0,\sin(1)]$  Since $|sin(x)|<|x|$ this will be a shorter interval than $[0,1]$.  We can apply the process repeated getting a succession of smaller intervals, which must either converge to the origin, or to an interval $[0,y]$ with $y=\sin y$.  But the only such $y$ is $y=0$, and the intervals shrink to the origin.
Thus, for any $\varepsilon>0$ there is $N$ such for $n>N,\ |\sin^{(n)}x|<\varepsilon$ for every $x\in \mathbb{R}$.  Thus, you are correct in saying that the graph will look like a straight line, in fact the $x$-axis.
