Denote $M : = \max(x_1, \ldots, x_N)$ and $m : = \min(x_1, \ldots, x_N)$. We have \begin{equation} \lambda_1 x_1 \leq \lambda_1 M, \quad \ldots, \quad \lambda_N x_N \leq \lambda_N M \end{equation} Applying R1904: Isotonicity of finite real summation, we therefore get \begin{equation} \sum_{n = 1}^N \lambda_n x_n \leq \sum_{n = 1}^N \lambda_n M = M \sum_{n = 1}^N \lambda_n = M \end{equation} In the other direction, we have \begin{equation} \lambda_1 x_1 \geq \lambda_1 m, \quad \ldots, \quad \lambda_N x_N \geq \lambda_N m \end{equation} whence we obtain \begin{equation} \sum_{n = 1}^N \lambda_n x_n \geq \sum_{n = 1}^N \lambda_n m = m \sum_{n = 1}^N \lambda_n = m \end{equation} $\square$