Number of Primes is Infinite/Proof 1

Theorem

The number of primes is infinite.

Proof

Aiming for a contradiction, suppose there exists a greatest prime number $N$.

Hence, let $S$ denote the finite set of all prime numbers.

Euclid's Theorem states that:

For any finite set of prime numbers, there exists a prime number not in that set.

Hence there exists $N'$ not in $S$.

If $N' < N$ then $N \in S$.

Hence:

$N' > N$

which contradicts the assumption the $N$ is the greatest prime number.

The result follows by Proof by Contradiction.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): indirect proof
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): indirect proof