KöMaL 2010 B4241

Let $p_1$=2, and for $n \ge$1 let $p_{n+1}$ denote the smallest prime factor of the number $np_1^{1!}p_2^{2!}\ldots p_n^{n!}+1$. Prove that every prime number occurs in the sequence $p_1,p_2,\dots\,.$