Here's a common proof of the infinitude of primes.
It's called a proof by contradiction.

Let's start by assuming P is the greatest prime number (that is, there are no primes greater than P).

Now let's let N = the product of all prime numbers less than or equal to P.
In other words, N = (2)(3)(5)(7)(11)(13)(17)....(P)
So, N is divisible by 2, 3, 5, 7, 11, 13, etc...

Now consider the value N+1 (a very big number)

We can see that, since N+1 is 1 greater than a multiple of 2, N+1 is NOT divisible by 2
Likewise, since N+1 is 1 greater than a multiple of 3, N+1 is NOT divisible by 3
And, since N+1 is 1 greater than a multiple of 5, N+1 is NOT divisible by 5
And, since N+1 is 1 greater than a multiple of 7, N+1 is NOT divisible by 7
.
.
.
And, since N+1 is 1 greater than a multiple of P, N+1 is NOT divisible by P.

So, N + 1 is not divisible by any of the prime numbers from 2 to P.
This means one of two things:

(1) N+1 is prime, which contradicts the statement that P is the greatest prime number.
(2) N+1 is not prime, which means it must be divisible by some prime number OTHER than the prime numbers from 2 to P, which is impossible.

Since the assumption that P is the greatest prime number leads us to two impossibilities, we can assume that there is no greatest prime number.