This is closer in spirit to the proof sketch given by Purcell. The key result is to show that a swimmer in a Stokes fluid does not depend on time. That is, a one cannot detect if a movie of a swimmer motion is slowed down, sped up, or reversed. The other results then are simple corollaries.
The stress tensor of the fluid is . 
Let  be a nonzero real constant. Suppose we have a swimming motion, then we can do the following scaling:and obtain another solution to the Stokes equation. That is, if we scale hydrostatic pressure, flow-velocity, and stress tensor all by , we still obtain a solution to the Stokes equation.
Since the motion is in the low Reynolds number regime, inertial forces are negligible, and the instantaneous total force and torque on the swimmer must both balance to zero. Since the instantaneous total force and torque on the swimmer is computed by integrating the stress tensor  over its surface, the instantaneous total force and torque increase by  as well, which are still zero.  
Thus, scaling both the swimmer's motion and the motion of the surrounding fluid scales by the same factor, we still obtain a motion that respects the Stokes equation.