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Let $a_n = \exp(-\cos(n))$. I need to find $\limsup_{n \to \infty}(a_n)$ and $\liminf_{n \to \infty}(a_n)$.

By applying the definition of $\limsup$ I got that: $$\limsup_{n \to \infty}(a_n) = \lim_{n \to \infty}(\sup_{n\ge m}(a_n)),$$ which should intuitively be $e$, and similarly the $\liminf$ should be $\frac{1}{e}$. How should we prove these statements?

Iamtrying
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  • This link should be helpful: https://math.stackexchange.com/questions/1110073/how-to-prove-the-set-cosn-mid-n-in-mathbbn-is-dense-in-1-1 – Umberto P. Jan 18 '22 at 14:56
  • Hint with lack of strictness : If $f$ is strictly increasing smooth function, then $$\limsup f(a_n)=f(\limsup a_n)$$ – MH.Lee Jan 18 '22 at 14:58

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