Question

prove that an electron cannot exist in the nucleus

Answer 1

Using quantum mechanics you could mathematically calculate that the probability of an electron existing inside the electron is nearly zero!.
Also using the Heisenberg's uncertainity principle if electron were confined to nucleus then the uncertainity in it's velocity exceeds the velocity of light which is well not so good

Answer 2

If the negatively charged particle electron exists in the positively charged nucleus then atom will collapse. That implies matter will collapse. But this is absurd. Hence electron cannot exist in nucleus