Prove that the sequence $\cos(n\pi/3)$ does not converge EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge...
$n=1$ to infinity of course.
So, I have a bit of trouble understanding the definition. Here's what I have so far:
I know it doesn't converge if there exists an $\varepsilon > 0$, where for ALL natural numbers $\mathbb{N}$, $|a_n - L| < \varepsilon$, for some $n\in\mathbb N$.
So I can make $\varepsilon > 1/2$, but then I'm not sure what to do. I know I need to prove this by way of contradiction using 2 cases, one where I evaluate $L$ as both greater than and equal the other where it's less than zero.

If the sequence converges, then all its subsequences have the same limit. However, the subsequence $\big( \cos \big( \frac{(6k+3) \pi}{3} \big) \big)_{k \in \mathbb{N}}$ converges to $-1$ and the subsequence $\big( \cos \big( \frac{(6k) \pi}{3} \big) \big)_{k \in \mathbb{N}}$ converges to $1$. Hence, the limit does not exist.
