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

Question

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 $0$ ?

Answer 1

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?

Answer 2

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.