  Based on the concept of extended particles recently introduced we perform a
Gedankenexperiment accelerating single electrons with photons of suitably low
frequency. Accounting for relativistic time dilation due to the acquired
velocity and in infinite repetition of single absorption processes it can be
shown that the kinetic energy in the infinite limit is equal to m_{e} c^{2}/2.
However, the inertial mass of the electron seems enhanced, and it can be
established that this enhancement is described by the relativistic mass effect.
It appears, therefore, that although there exists a singularity in interactions
- the frequency required to accelerate the particle near the limit of c becomes
infinite - the energy of the particle itself approaches a finite limit.
Comparing with calculations of the Lamb-shift by Bethe this result seems to
provide the ultimate justification for the renormalization procedures employed.
