I found an interesting and very clear explanation as to why the prime numbers must be infinite. It's not very mathematical or rigorous, but it was definitely easy to understand, and got me thinking about things in a slightly different way. Here is the source: http://gwydir.demon.co.uk/jo/numbers/interest/infinity.htm.

First, assume that the set of all prime integers is finite. This means we can write down every single prime number.

Now, if we could write every prime number, then we could, theoretically, multiply them all together. This result will have all the known primes as factors.

Now add one to it. This new number will have none of the known primes as factors. So either it will be a new prime, or it will have at least one new prime as a factor. This implies that it is impossible to write down every single prime! So the set of prime numbers must be infinite.