Generating Function for Elementary Symmetric Function/Proof 3

Proof
We have by definition of generating function that:
 * $G \left({z}\right) = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$

We have that:
 * $a_0 = 1$

Suppose $n=1$.

Let $G \left({z}\right)$ be the generating function for $\left\langle{a_m}\right\rangle$ under this condition.

Then:
 * $1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le 1$

can be fulfilled by only one set $\left\{ {j_1, j_2, \ldots, j_m}\right\}$, that is:
 * $j_1 = 1$

Thus in this case:

Then by Product of Generating Functions, it follows that:


 * $G \left({z}\right) = \left({1 + x_1 z}\right) \left({1 + x_{2} z}\right) \cdots \left({1 + x_n z}\right)$