The proof of the statement "At least one of the renormalization constants in electrodynamics is infinite" is examined in the light of perturbation theory and the gauge invariance of electrodynamics. The essential result used to derive the statement is found not to reproduce perturbation theory at least in a simple way. On the basis of gauge considerations a conjecture is proposed which provides a modified essential result and which is found to reproduce perturbation theory. Even if the modified result could be rigorously established, it would not lead to the statement that any gauge-independent quantity is infinite. In fact, the combined results would establish only the statement that the use of gauges where the exact electron "wave function" relative to the "wave functions" for a free electron is a constant, is not consistent.