Generating Function for Elementary Symmetric Function/Proof 3

Proof
We have by definition of generating function that:
 * $\map G z = \ds \sum_{n \mathop \ge 0} a_n z^n$

We have that:
 * $a_0 = 1$

Suppose $n = 1$.

Let $\map G z$ be the generating function for $\sequence {a_m}$ 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 $\set {j_1, j_2, \ldots, j_m}$, that is:
 * $j_1 = 1$

Thus in this case:

Then by Product of Generating Functions, it follows that:


 * $\map G z = \paren {1 + x_1 z} \paren {1 + x_2 z} \cdots \paren {1 + x_n z}$