What is $\limsup\limits_{n\to\infty} \cos (n)$, when $n$ is a natural number? I think the answer should be $1$, but am having some difficulties proving it. I can't seem to show that, for any $\delta$ and $n > m$, $|n - k(2\pi)| &lt \delta$. Is there another approach to this or is there something I'm missing?

You are on the right track.  If $|n-2\pi k|&lt\delta$ then $|\frac{n}{k}-2\pi|&lt\frac \delta k$. So $\frac{n}{k}$ must be a "good" approximation for $2\pi$ to even have a chance.
Then it depends on what you know about rational approximations of irrational numbers.  Do you know about continued fractions?
