I was wondering if it was possible to formalize the following. If we let $f_n(x)$ denote the sequence of function so that $f_1(x)=\sin(x)$, $f_2(x)=\sin(\sin(x))$, $f_3(x)=\sin(\sin(\sin(x)))$ and so on. Does this sequence of functions have a limit? Is there a theory that could explain the behaviour of repeating trigonometric functions?
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Whoops, tried searching. Thank you @MartinR – Lundborg Nov 02 '15 at 18:49
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When I was a kid, we used to take one of those clumsy calculators with big buttons, enter some arbitrary number, press down the "Sin" button with a brick and leave it for an hour or so, in an attempt to answer the same question. – Ivan Neretin Nov 02 '15 at 18:50
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In fact, this is a neat application of the Banach fixed point theorem. It turns out that $$ \lim_{n \to \infty}f_n(x) = 0 $$ So that the sequence of functions $f_n$ converge pointwise to $f(x) = 0$. In fact, it can be shown that this convergence is uniform.
A slightly more interesting case is that of $f_1(x) = \cos(x)$.
Ben Grossmann
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HINT: $|\sin x|<|x|$ for $x\neq0$, hence the limit exists. Can it be not equal to 0?
Przemysław Scherwentke
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