# Prime numbers dividing infinitely many numbers in an sequence

Here I found a question:

Show that every prime not equal to $$2$$ or $$5$$ divides infinitely many
of the numbers $$1,11,111,1111,dots$$ etc.

which is partly solved here Prime numbers divide an element from a set.

From this the following conjecture seems reasonable:

Given any finite set $$S={q_1,dots,q_k}$$ of $$k$$ primes, then any
prime $$pnotin S$$ divides infinitely many of the numbers
$$a_1,a_2,a_3dots$$, where $$a_1=1$$ and $$a_{n+1}=1+a_nprod q_i$$.

Can this be proved?