I don't think your proof is quite what's necessary. I think you should show that
Different books sometimes define things differently, but if you can prove that one of the one-sided limits is infinite, then the limit does not exist. So how would you go about proving this? The definition of this limit is as follows:
if and only if
So, you need to prove that
In answer to your sequence question, in this context, n is just a number you've defined in your proof.