Euclid proved that there are an infinite number numbers. He did this by first considering the possibility that there are a finite number of primes.
He knew that by definition all non-prime numbers were expressible by multiply together different combinations of prime numbers. Then he considered multiply together all the known prime numbers and adding 1. He realised that this number could not be divisible by any of the known primes because dividing by any of the known prime numbers would leave a remainder of 1. This number would therefore either have to be prime itself, or be divisible by another prime number that was not known.
In this way he was able to show that however many prime numbers were known, it would always be possible to show that there are more.