I was just wondering if my method for finding lim sup and lim inf for sequences is correct. It's different to how we do it in class, but I asked my lecturer about my way and he said that it looks fine but he'd think about it and get back to me. Essentially what I do is first take the sequence to infinity and then find its sup and inf at infinity, instead of finding the sup/inf first and then taking this to infinity. Here is an example to clear up what I'm saying:
Let the sequence x_n = (-1)^n + n/(n+1). What I first do is find the limit as n tends to infinity for each term. In this case, the first term alternates between 1 and -1, and the second term tends to 1. Once this is done I find the supremum and infimum. Therefore,
lim sup x_n = 1 + 1 = 2 and lim inf x_n = -1 + 1 = 0.
Is my method fine, or could there be a case where it would not work?