Why don’t electrons fall into the nucleus of an atom?

Question

The nuclei of atoms are composed of protons and neutrons. The proton has a positive charge, and the neutron a neutral charge. Shells of electrons outside the nucleus in the atom orbit around the nucleus. Here is my question: If the protons in the nucleus possess a rival charge to the electron, why doesn’t the electron become attracted to the proton(s) in the nucleus and hence fall “into” the atom?

Answer 1

The Sun attracts the Earth, so why don't we fall into the Sun? There is a different problem for the atom, though. When electrons accelerate, they give off energy as photons. Orbiting in a circle is still form of acceleration, so the electron should lose all of its orbital energy and then fall into the nucleus. This doesn't happen, so we know that the orbiting electron picture is wrong. Quantum mechanics tells us the electrons act like waves around the nucleus.