Is the following any clearer? (Read the subscripts carefully!)

$$\begin{array}{ll}
 f_1(y_1) + f_2(y_2) + f_3(y_3) + \cdots + f_n(y_n)  &\leq  \\
 f_1(y_2) + f_2(y_2) + f_3(y_3) + \cdots + f_n(y_n) & \leq \\
 f_1(y_3) + f_2(y_3) + f_3(y_3) + \cdots + f_n(y_n) & \leq \\
\qquad \qquad \qquad \vdots & \leq \\
 f_1(y_n) + f_2(y_n) + f_3(y_n) + \cdots + f_n(y_n) & \leq \\
 f_1(x) + f_2(x) + f_3(x) + \cdots + f_n(x). &\\
\end{array}$$

I don't think this is different from the induction proof in any serious way, but sometimes "$\cdots$" is more intuitive than a formal induction.