Proof is evidence that shows something is definitely true or that something really happened. When a detective finds proof that someone committed a crime, they've discovered evidence so strong it removes all reasonable doubt. When you show proof of your age to get a library card, you're providing a document like a birth certificate that confirms when you were born.

In mathematics, proof has a special meaning. A mathematical proof is a logical argument that shows why something must be true, using reasoning so careful that no one can find a flaw in it. When mathematicians prove a theorem, they demonstrate it will work every single time, not just in the cases they tested. For thousands of years, Euclid's proofs about geometry have held up because they rely on pure logic rather than just examples.

Scientists talk about proof differently. They gather evidence and run experiments, but they know new discoveries might change their understanding. That's why scientists usually say their findings “support” a theory rather than prove it absolutely.

The phrase the proof is in the pudding means you can't judge something until you've tested or experienced it yourself.