I know Euclid's proof of there being infinite number of primes. I want to let my brother (age 15) arrive at that proof by himself. He knows definition of a prime number (number divisible only by 1 and itself).
First when I asked him how many prime numbers are there, he said there must be an infinite number of them. Why? Because it's not possible to know which one is the largest. Why? ... blank.
Then I told him that since larger a number becomes, more numbers are there below it. So chances of a larger number
N being divisible by at least one of the
N-2 numbers increases as
N-2 larges with
N. He agreed with this logic.
Then I asked him if he could generate a prime number larger than any two given given primes (and a third constant) using some simple function of them. I was trying to make him arrive at $prime1*prime2 + 1$ formula. I gave him two primes 2 and 5 and asked to think of some examples. He summed them -> 7 is a prime. Then I asked him to use 1 also, he said 251 is a prime. Then he got bored.
What approach can I use to intuitively arrive at the proof?