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.