Prove that for $x_i > 0$ and $x_i$ distinct such that $x_1 < x_2... < x_{2n+1}$,
$\displaystyle\sum_{i = 1}^{2n + 1} (-1)^{i+1} x_i \leq \left(\sum_{i = 1}^{2n+1} (-1)^{i+1} (x_i)^n\right)^{\frac{1}{n}}$.
Prove that for $x_i > 0$ and $x_i$ distinct such that $x_1 < x_2... < x_{2n+1}$,
$\displaystyle\sum_{i = 1}^{2n + 1} (-1)^{i+1} x_i \leq \left(\sum_{i = 1}^{2n+1} (-1)^{i+1} (x_i)^n\right)^{\frac{1}{n}}$.