Elementary Symmetric Function/Examples/m Greater than n

Example of Elementary Symmetric Function: $m > n$
Let $\map {e_m} {\set {x_1, x_2, \ldots, x_n} }$ be an elementary symmetric function in $m$ variables of degree $n$.

Let $m > n$.

Then:

Proof
By definition:

However, there does not exist a set of $m$ integers which is a subset of $\set {1, 2, \ldots, n}$.

That is, the summation $\displaystyle \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n}$ is vacuous.

Hence the result.