Assume the Opposite

Proof by contradiction is the most elegant weapon in mathematics. The strategy: assume the opposite of what you want to prove, then follow the logic until it collapses into absurdity. If assuming "X is false" leads to nonsense, then X must be true.

This is not a trick. It is a rigorous method that has settled some of the deepest questions in mathematics. Euclid proved there are infinitely many prime numbers back in 300 BC, and his argument has not been improved in over 2,000 years. We will recast it as a contradiction shortly, though his original version was a direct one.

The real power is that it works precisely when direct proofs fail. You do not need to build the thing you are proving. You only need to show that its nonexistence leads to an impossibility.