Word growth of groups - Z^d has polynomial growth degree d

Hi,

I am trying to prove that Z^{d }has polynomial growth rate degree d. This seems to be a standard fact but I haven't come across a proof.

My attempt is:

Let {x_{1 }, ... , x_{d}} be the standard generating set.

Then any word of length less than or equal to r is a product of the form (x_1)^(s_1)...(x_d)^(s_d) , where the s_i are integers and the sum of the |s_i| is less than or equal to r. So we can view this as choosing d words in Z, each of length less than or equal to r. Hence the number of words length less than or equal to r is less than or equal to (2r+1)^d. (The growth rate of Z is 2r+1 - proved by induction)

So now we need to bound this below by a polynomial of degree d. My attempt at doing this is as follows: the number of words of length less or equal to d.r is greater than or equal to (the number of words in Z of length less or equal to r)^d , ie (2r+1)^d . So we've bounded below the growth rate at d.r for each r by a polynomial in r of degree d.

Is this sufficient to conclude that the growth rate is polynomial degree d?

Is there a proof of this fact which gives a formula for the growth function of Z^d (with respect to the standard basis)?

Thanks in advance for any help!