Wilson’s Theorem

Wilson’s theorem states that for any prime integer $p$, $(p-1)! \equiv -1 \pmod{p}$. I wrote this on the board at the beginning of lecture as a warm-up for my students. But when asked to give a proof, I became dissatisfied with my reasoning when a student said he didn’t understand the argument, and I could not manage to explain it. I believe this is a correct proof.