Comment by tgarrett

If the electron starts with zero energy at infinity (e.g. a parabolic orbit, a natural default assumption), and some of the potential energy is converted into free EM radiation due to acceleration of the electron as it is falling down the potential well, then it will become bound to the proton. My reading of phkahler's original statement is that the electron will wind up going faster than the speed of light (which is incorrect, due to gamma) due to falling down the potential well, and not due to having non-zero kinetic energy at infinity...