Generating Function for Elementary Symmetric Function/Proof 2

Proof
Apply mathematical induction on $n$.

Let $\map P n$ be the statement:

Basis for the induction:

Set $U = \set {x_1}$ for $n = 1$.

Expand the formal series:

Then $\map P 1$ holds.

Induction step:

Assume $\map P n$ holds.

Let's prove $\map P {n + 1}$ holds.

The induction step uses a recursion relation:

Let $\map G z$ be defined by statement $\map P n$.

Let $\map {G^*} z$ be defined by statement $\map P {n + 1}$.

Then:

Then $\map P {n + 1}$ holds, completing the induction.