Euclid's Infinite Number of Primes

Euclid. There are an infinite Number of Primes

Proof $p_1=2,p_2=3,\dots$ be the sequence of primes in ascending order, and let $p_n$ be the last prime. Now lets construct the following number $n=p_1{p}_2\cdots p_n+1$

Suppose for the sake of contradiction that there are a finite number of primes. Let