Applying R4228: Real ordering is compatible with addition repeatedly, we have \begin{equation} \begin{split} \sum_{n = 1}^N x_n & = x_1 + x_2 + x_3 + \cdots + x_{N - 2} + x_{N - 1} + x_N \\ & \leq y_1 + x_2 + x_3 + \cdots + x_{N - 2} + x_{N - 1} + x_N \\ & \leq y_1 + y_2 + x_3 + \cdots + x_{N - 2} + x_{N - 1} + x_N \\ & \leq y_1 + y_2 + y_3 + \cdots + x_{N - 2} + x_{N - 1} + x_N \\ & \; \; \vdots \\ & \leq y_1 + y_2 + y_3 + \cdots + y_{N - 2} + x_{N - 1} + x_N \\ & \leq y_1 + y_2 + y_3 + \cdots + x_{N - 2} + y_{N - 1} + x_N \\ & \leq y_1 + y_2 + y_3 + \cdots + x_{N - 2} + x_{N - 1} + y_N = \sum_{n = 1}^N y_n \end{split} \end{equation}