When proving that a certain sequence does not converge, is it enough to show that there exist 2 sub-sequences, one that converges to a limit $L$, and the other diverging (increases without bound for example) to prove this, or must I show that there exist two different sub-limits? (I'm using the rule that every sub-sequence of a convergent sequence converges to the same limit.)
For example:
When proving that $a_n = \cos\left(\frac{\pi n^2} {2n+3}\right)$ does not converge, I found two sub-sequences, $b_n$ and $c_n$, where $b_k = \frac1 k$ and $c_k = \sqrt{k}$, in which $a_{b_k}$ converges to $0$ and $a_{c_k}$ diverges.
EDIT
Just realized the above aren't indices at all... since $k$ must be integers. I've tried subs-sequences of even and odd integer indices for $k$ but don't see a pattern for the function.