Generating Function for Binomial Coefficients

Theorem
Let $\left \langle {a_n}\right \rangle$ be the sequence defined as:
 * $\displaystyle \forall n \in \N: a_n = \begin{cases}

\binom m n & : n = 0, 1, 2, \ldots, m \\ 0 & : \text{otherwise}\end{cases}$

where $\displaystyle \binom m n$ denotes a binomial coefficient.

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $\displaystyle G \left({z}\right) = \sum_{n \mathop = 0}^m \binom m n z^n = \left({1 + z}\right)^m$

Proof
The result follows from the definition of a generating function.