For any integer $n\geq 1$, define $\sin_n=\sin\circ ... \circ \sin$ ($n$ times). Prove that $\lim_{x\to 0}\frac{\sin_nx}{x}=1$ for all $n\geq 1$ I got this problem:
For any integer $n\geq 1$, define $\sin_n=\sin\circ ... \circ \sin$ ($n$ times). Prove that $\lim_{x\to 0}\frac{\sin_nx}{x}=1$ for all $n\geq 1$.
Some hints will be appreciated.

Hint: this is true for $n=1$. Note that $\sin_0x=x$ and therefore it gives you $$\lim_{x\to0}\frac{\sin_1 x}{\sin_0 x}=1.$$
This is probably enough to figure out that you should consider
$$\lim_{x\to0}\frac{\sin_{n}x}{\sin_{n-1}x}.$$
