Prime numbers/Infinity/Fact/Proof

Proof

We assume that the set of all prime numbers is finite, say $\{p_1,p_2,\dots ,p_r\}$ is a complete list of all prime numbers. We consider the number

$N=p_1\cdot p_2\cdot p_3\cdots p_r+1.$

This number can not be divided by any of the prime numbers $p_i$, since the reminder of $N$ by division through $p_i$ is always $1$. Hence, the prime factors of $N$, which exist due to fact, are not contained in the given set. This is a contradiction.

To fact