Lemma 2

$p \neq 0$ and $\sigma p = (-1)^{\nu(\sigma)}$ for $\sigma \in \mathfrak{S}_n$.

$p(1, 2, \ldots, n) = \prod_{i < j} (j - i) \neq 0$ and hence $p \neq 0$. On the other hand, if $\sigma \in \mathfrak{S}_n$, then

Let C be the set of ordered pairs $(i, j)$ such that $1 \leq i \leq n,\ 1 \leq j \leq n,\ i < j$. A permutation $\theta$ is defined on C by setting $\theta(i, j) = (\sigma(i), \sigma(j))$ if $(i, j)$ is not an inversion, $\theta(i, j) = (\sigma(j), \sigma(i))$ if $(i, j)$ is an inversion. This implies $\sigma p = (-1)^{\nu(\sigma)} p$.