Oh, finitely generated - of course! (slaps head).

No, not finitely generated! Infinitely generated, but every element in N is a finite linear combination of this infinite basis. (so slap it again (Happy) ). From what book/site is this question? I kindda remember a very similar question, but cannot place it.
Now, if

, given k>=1 I have

,

so that

as per the hint.

Hmmm...I think it'd rather be , so that: , and thus: It's easy to see that the above is true, mutandis mutandis, if instead powers of 2 we choose powers of 3. Now, as , then Tonio
As g(N)=0, taking g of both sides gives

for any k>=1.

In particular,

so, upon subtracting:

.

This would seem to imply x=(0,0,..) or g=0; in either case g(x)=0.

But this can't be right - I could have deduced as much by not bothering with the powers of 2. What am I missing?

Thanks again