prove that an electron cannot exist in the nucleus
Stacey Warren - Expert brainly.com
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when electron exist in the nucleus then proton and electron collapsed. So atom could not exist. unless atom the whole universe couldn't exist. This is the theoretical fact. But I don't know the mathematical proof.
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 :P.