\displaystyle $\begin{align*}
x_1 = 1 \cdot x_1 = \left( \sum_{i = 1}^n p_i \right) x_1 = \sum_{i = 1}^n p_i x_1 \leq \mathbb{E}X = \sum_{i = 1}^n p_i x_i \leq \sum_{i = 1}^n p_i x_n = \left( \sum_{i = 1}^n p_i \right) x_n = 1 \cdot x_n = x_n
\end{align*}$