Euclid's Theorem/Corollary 2/Proof 1
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Corollary to Euclid's Theorem
There is no largest prime number.
Proof
Let $\mathbb P$ be the set of all prime numbers.
Aiming for a contradiction, suppose there exists a largest prime number $p_m$.
Then:
- $\mathbb P \subseteq \closedint 1 {p_m} = \set {1, 2, \ldots, p_m}$
and so $\mathbb P$ is a finite set.
By Euclid's Theorem, there exists a prime number $q$ such that $q \notin \mathbb P$.
But that means $q \notin \closedint 1 {p_m}$.
That is, $q > p_m$.
So $p_m$ is not the largest prime number after all.
Hence the result, by Proof by Contradiction.
$\blacksquare$