Please prove that $(p!+1)$ is a prime, where $p$ is a prime. I got Euler's proof for the Question:
Prove that there is no largest prime.
But one of the solutions on internet gives this proof by Archimedes:

Please help me to understand the proof.

Note that $p!+1\gt p$. By assumption $p$ is the largest prime, so $p!+1$ cannot be prime. Hence it has a prime divisor $q|p!+1$. But $q\le p$, so $q|p!$. From this we conclude $q|p!+1-p!=1$, which is absurd. If the assumption that there is a largest prime leads to a contradiction, then the assumption must be false and there is no largest prime.
