Proposition 5

— The set of prime numbers is infinite.

Indeed, given an arbitrary finite family $(p_i)$ ($1 \leq i \leq n$) of distinct prime numbers, any prime divisor $q$ of the number $\left( \prod_{i=1}^n p_i \right) + 1$ (which is $> 1$) is distinct from all the $p_i$, for otherwise it would divide 1.