Prove that an electron in a hydrogen atom doesn't emit radiation

You have your "prove" in the wrong place. The way to prove that ground-state electrons in hydrogen atoms don't emit radiation is the following:

  1. Construct a sample of ground-state neutral hydrogen atoms.
  2. Place this sample near a detector which is sensitive to the sort of EM radiation you expect.
  3. Die of old age waiting for a signal, because ground-state hydrogen doesn't emit radiation.

This experimental evidence demonstrates that classical electromagnetism, in which accelerating charges emit radiation, does not describe the hydrogen atom.

The existence of hydrogen atoms is enough to demonstrate that the electrons don't emit radiation.

If they did, that energy would have to come from somewhere. The only place it could come from would be a reduction of orbital radius until the electron finally reaches the nucleus.

If you accept that electrodynamics applies, then you have to accept that atoms cannot exist - since they do, electrodynamics must not be the whole story.

Because of its wave nature, the electron in its ground state is actually smeared symmetrically about the proton (ignoring spin-spin effects), and spherically symmetric charge distributions do not radiate (there's no special direction). Accelerated charges do not always radiate em radiation. See also How to find the magnetic field due to a revolving electron of hydrogen atom in first orbit