Brain Teaser - Can you complete this proof that everyone (yes, including you) ought to know?

In every subject there are certain things that should be known by the general public. There are certain paintings, novels and poems that an educated person should be familiar with, though this knowledge may not make you wealthier or improve your love life. I say the same is true of mathematical proofs.

This brain teaser is based on a very short and very sweet proof due to Euclid that shows there are infinitely many prime numbers. I will start you off. See if you can finish and thereby receive an aha! moment.

The proof is usually presented as indirect, but I think that is an unnecessary complication. The proof is constructive, meaning that you can use it, at least in theory, to find a prime number larger than any given number.

Given a number n, consider all the primes p1,...,pk that are less than or equal to n. Actually finding these primes is a bit of work, but it can always be done in a finite amount of time. Now multiply all these primes together and add 1. That is, create the number m=p1*p2*...*pk + 1. Why does this number m show that there is a prime greater than n?

Please don’t spoil if for others. You can send me a PM with the answer or, if you prefer, explain why you do not see any way that knowing this proof will make you a better person. I will give the answer in two days.