For exercise, use the fact that for all integers
n, n! =n(n − 1)… 3 • 2 • 1.
Exercise
An alternative proof of the infinitude of the prime numbers begins as follows:
Proof: Suppose there are only finitely many prime numbers. Then one is the largest. Call it p. Let M = p! + 1. We will show that there is a prime number q such that q > p. Complete this proof.
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