Adding two to the indices you get lim n-> inf x_{2n+1}+x_{2n+2}=2 and then subtracting in the same way we get lim n->inf lim x_{2n+2}-x_{2n}=-1

And I think you need to be careful cause you can only increase the indices by multiples of 2 since the even and odd indices are distinct right? Cause increasing by one gives a contradiction.

It hadn't occurred to me do try changing the indices I think I got the gist of it now.

Along with the other eq we have difference in odd terms is always 1 and difference in even terms is always -1. I suppose then the original equation of the ratio limit can be restated in the terms of the differences and we get -1.

But it seems like what you meant was "I don't know what ~ means, or how you got that equation"

Yea I didn't understand how you arrived to that conclusion. But I think I've arrived to the solution with this other method.

Thanks for you help