Why Not Just Prove It Directly?

A direct proof starts from known facts and builds the conclusion without assuming the opposite. Euclid's original prime argument can be read that way: take any finite list of primes, build a number from it, and show some prime lies outside the list.

Contradiction is a different framing. It assumes the list is complete, then shows that assumption breaks. Both routes prove there is no finite list of all primes.

Notice what the proof never does: it never finds a "last" prime or builds the full list. It only shows that a finite list is impossible. That is the signature of a contradiction proof — it rules out the alternative instead of constructing the answer, which is why it can reach conclusions a direct proof cannot touch.