<s>[INST] Find limit supremum from 3 sequence theorem Find limit $$\limsup_{n\rightarrow \infty} \sqrt[n]{\left| \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}  \right) \right)^p\right |} $$
dependency on value of $p$
I think that $p$ doesn't matter there (by doesn't matter I mean that I can compute limit without knowledge about p and on the end power result to p) because it is just const value so I can write that as
$$\limsup_{n\rightarrow \infty} \left(\sqrt[n]{\left| 2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}    \right)\right|} \right)^p$$ and just calculate
$$\limsup_{n\rightarrow \infty} \sqrt[n]{\left| 2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}    \right)\right|} $$
I suspect that this sequence has normal $lim$ so probably
$$\limsup_{n\rightarrow \infty} \sqrt[n]{\left| 2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}    \right)\right|}  = \lim_{n\rightarrow \infty} \sqrt[n]{\left| 2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}    \right)\right|} $$
Ok now I want to use three sequence theorem:
$$ ... \le \sqrt[n]{\left| 2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n}    \right)\right|} \le \sqrt[n]{2} $$
But I don't know how to estimate that from below...
 [/INST] Hint
Compose Taylor series for large values of $n$ to get 
$$2-2 \cos\left(\frac{1}{n}\right)-\frac 1n{\sin \left(\sin \left(\frac{1}{n}\right)\right)}=\frac{1}{4 n^4}-\frac{7}{72 n^6}+O\left(\frac{1}{n^8}\right)$$
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