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A sequence $\left({a_{n}}\right)_{n\in\mathbb{N}}$ is contractive iff there exists a constant $c$, with $0, such that:

$|{a_{n+2}-a_{n+1}}|\leqslant{c}\,|{a_{n+1}-a_{n}}|$, for all $n\in\mathbb{N}$.

Examine if the sequence $a_{n}=({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-{\rm{times}}}})({n})$, $n\in\mathbb{N}$, is contractive.

  • 1
    See [this question](http://math.stackexchange.com/questions/45283/lim-n-to-infty-sin-sin-sin-n).2012-01-31
  • 1
    As you see from the answer below the limit of $a_{n}$ doesn't help.2012-02-02

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