Monotone Additive Function is Linear/Proof 2

Proof
We use a Proof by Contraposition.

To that end, suppose $f$ is not linear.

We know that Graph of Nonlinear Additive Function is Dense in the Plane.

Therefore $f$ is not bounded on any nonempty open interval.

But then $f$ is certainly not monotone.

Hence, by Rule of Transposition, if $f$ is monotone, then it is linear.