Proof that there are infinite prime numbers.
Reductio ad absurdum:
Say there are a finite amount of prime numbers. Name them a, b, c, d, e, ....... x, y, z.
Consider the number (a*b*c*d*e*...*x*y*z)+1
This number has no factor among the identified primes. Therefore, either a prime number was left off of the list, or this number itself is a prime.
This process continues indefinitely. Therefore, there is not a finite number of primes, so ther emust be infinite prime numbers.