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}