Base b and generating functions

 Not a puzzle, just a quick observation noting the power of generating functions.

Say $b \ge 2$ is a positive integer. We give a quick proof using generating functions that every positive integer can be written uniquely in base $b$ with digits $0, 1, \dots, b-1$.

Let $n \ge 1$. Consider 

$$ G(x) = \prod_{k=0}^{n} (1 + x^{b^k} + x^{2b^k} + \dots + x^{(b-1)b^k})$$

Observe that the coefficient of $x^N$ shows the numbers of ways of writing $N$ is base $b$.

Now if $u = x^{b^k}$, then

$$ (1 + x^{b^k} + x^{2b^k} + \dots + x^{(b-1)b^k}) = (1 + u + u^2 + \dots u^{b-1})  = \frac {1 - u^b}{1-u}$$

Thus

$$G(x) = \frac{1-x^b}{1-x}\frac{1-x^{b^2}}{1-x^b}\frac{1-x^{b^3}}{1-x^{b^2}}\dots\frac{1-x^{b^{n+1}}}{1-x^{b^n}}$$

This telescopes to

$$ G(x) = \frac{1 - x^{b^{n+1}}}{1-x}$$

$$ = 1 + x + x^2 + \dots + x^{b^{n+1} - 1}$$

Thus every integer $N$ in the range $1$ to $b^{n+1} - 1$, can be written in base $b$ using at-most $n+1$ digits, taken from $0,1,2, \dots, b-1$. Since the coefficient of $x^N$ is $1$, the representation is unique.