Are there arbitrarily long arithmetic progressions in which all the prime factors of all the terms are at most $N$, for some $N$? Assume all the terms are positive and the sequence of terms is increasing.

I have proved that no such infinite sequence exists. Note the $N$ may vary from AP to AP. For infinite sequences let $\{a+nd\}_{n\ge 0}$ be the AP. $\text{gcd}(a,d)=s$ then $a+nd=s(x+ny)$ for some $x,y$ with $\text{gcd}(x,y)=1$ then by Dirichlet's theorem $\{x+ny\}_{n\ge 0}$ has infinitely many primes. Thus we have the prime factors of the sequence is unbounded and hence done. But I was thinking about this claim but nothing came in mind. Could someone help? Thanks a lot.