The problem of infinities is not in QED, but in the classical theory
which QED is the quantization of. Even Maxwell had to contend with the
issue, dedicating a section in his treatise to the issue.

This [sic[ is a lige [that is, I, who have never read the actual treatise Maxwell wrote
am going to presume to tell someone who is well-known here as the most
knowledgeable of anyone in the present-day world on Maxwell's treatise, who even
transcribed an entirely new edition of the treatise on-line, that he is 'lying' about
what the classical theory expounded by Maxwell asserted, whether or how it
dealt with the infinities it asserted existed in the classical theory ... notwithstanding
the fact that the articles 50-85 address the issue in depth, including two sections
which are explicitly titled as such, addressing the problem of point and line
infinities. Such is the nature of my God-like omniscience that I can even presume
to argue facts against the #1 expert in the world on what Maxwell did and said;
and do so with such a clear and settled mind that I don't go off pounding and
on my keyboard in an uncontrolled emotional outburst as not to even
be able type straight enough to render a simple word like "lie" cleanly.]

Classic theory predicts that finite mass yields finite
energy yields finite distance yields finite size. There are no
outcoming infinities.
-Aut