# Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0$?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0$$ then I'm done. It's monotonic and decreasing and bounded by 0 and 1 respectively, so it converges, though is it truly $0$ ?

• Hint: what solutions are there to the equation x = sin(x) – John Apr 3 '14 at 20:19
• yup, now I get it... wonder if that's what they expect of us. – user7610 Apr 3 '14 at 20:27
• – Rustyn Apr 3 '14 at 22:15

Hint: From what you stated, you've proven that it converges to some limit, call it $L$. You know that $0 \leq L \leq 1$. Do you see an identity that $L$ satisfies with respect to sine?
• I treated this problem as a fixed point and tried to prove it via banach but it's seems like $\sin(x)$ is not a contraction map on $[0,1]$. Or not? – rlartiga Apr 3 '14 at 21:14
This is a recursive function. Start with a definition: $$\sin(x)=x$$ This is already a dead giveaway, the only place where this holds true is at $x=0$. This is the answer, it's this simple. The $\sin(1)$ in the middle is a little misleading.
As you can see here, it's slowly getting down to $0$ as you approach infinity. Also note that the test one ($\sin 1337$) isn't much different from the original because this recursive function will hold true for whatever is inside at the end.