Prove the halting theorem
The halting theorem states that a program will never be able to determine, by itself, whether it will ever halt or not.
Proof:
We assume that there exists a program P that can determine, by itself, whether it will halt or not.
Let us consider a program Q that takes an input X and runs P with X as an input.
If P determines that it will halt, then Q will halt.
If P determines that it will not halt, then Q will also not halt.
But this contradicts the assumption that P can determine, by itself, whether it will halt or not.