[Note: there are many parts to the original question. This is only part of it].

I am not sure how to proceed here. Firstly I know that the above function is a translation. I also know that if delta has an infinite order then delta^{n} cannot equal the identity for any n which is an element of Z.

Now I can see that x delta^{n} = (x+n, (-y)^{n}) and that delta^{n} will never return to (x,y).

But I am not sure how to prove this. If anything is unclear, please ask and I will try to clarify my question.

Do I first prove by induction on n that

x delta^{n} = x + n for all x, n > 0

and if n > 0 then x + n is not equal to x so delta^{n} does not equal the identity?