Primes $=_m 1 $. For any positive integer m, prove that arithmetic progression
$$1 +m, 1 + 2m, 1 + 3m, ... $$
contains infinitely many primes.
How can we prove that it suffices to show that for all $m$ positive, the arithmetic progression contains at least one prime?
Answered by @RobertIsrael in the comments.
Given that for all positive integers $n$ there is a prime in the arithmetic progression $$ S_n=\{1+n,1+2n,1+3n,\ldots\} $$
then $S_m$ must contain at least one prime. But it cannot contain a largest prime, since for any $N$ the subsequence $S_{Nm}\subseteq S_m$ must contain a prime larger than $N$, thus it must contain infinitely many.