Since $f$ is proportionally bounded, there exists $C > 0$ such that $\Vert f(x) \Vert_M \leq C \Vert x \Vert_N$ for every $x \in N$. If $x, y \in N$, then linearity implies \begin{equation} \Vert f(x) - f(y) \Vert_M = \Vert f(x - y) \Vert_M \leq C \Vert x - y \Vert_N \end{equation} Since $x, y \in N$ were arbitrary, the claim follows. $\square$