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: $\sin(x)\sim x$ then, $\sin(\sin(x))\sim\sin x\sim x$ then $\sin_n(x)\sim\sin_{n-1}(x)\sim...\sim x$. I let you conclude.
