Question

Real analysis: Prove that the limit (approaching infinity) of n(-1)^n does not exist

Answer 1

So far I have:
Assume \[\lim_{n \rightarrow \infty} n(-1)^{n}=L\]
Then \[\exists N s.t. n >N \implies \left| (-1)^{n} n\right|<1\]

Answer 2

Do I use two cases, one where n is even and one where n is odd?

Answer 3

you have to first know how to negate the definition

Answer 4

if the limit is L the for all \(\epsilon>0\) \(\exists N\) such that for all \(n>N\) \(|a_n-L|<\epsilon\)

Answer 5

to negate that, you have to say \(\exists e>0\) such that \(\forall N\) there exists \(n>N\) with \(|a_n-L|>\epsilon\)

Answer 6

this should be pretty easy in this case, because no matter what \(L\) is can make \((-1)^nn\) arbitrarily large (or small) so that irrespective of \(L\) you can make \(|(-1)^nn-L|>1\) if you pick \(\epsilon=1\) for example

Answer 7

Ok, thanks I think I can take it from there. I will ask if I get stuck.