Question

why electron does not found inside the nucleus. proof mathematically

Answer 1

The electron can exist inside a nucleus. It just can’t stay there every long as a free electron because it would have to radiate way its orbital energy, and it cannot do that because there is no stable energy level for it. Essentially the Schrödinger equation solves for two degrees of freedom, and the 1s state puts half a quantum of action in the radial and angular motion. It cannot radiate without losing/changing one quantum off action, so it cannot do that. If it had enough energy, it could convert a proton to a neutron, but it does not usually have enough energy. (Electron capture is possible with some rather unstable nuclei.) So the electron can get into the nucleus, but it usually comes out again very quickly.