Proof techniques: direct, contradiction and induction

Around 300 BC, Book IX of Euclid's Elements proved that the primes never run out. The argument fits in a few lines: take any finite collection of primes, multiply them together and add one. The result leaves remainder 1 when divided by each of them, so its prime factors are new primes that were not in the collection.

No amount of checking could have established that. Testing the first billion numbers says nothing about the next one. A proof does, and the same skill is what lets you claim that a loop terminates or an algorithm is correct for every input.