Primes of form k*n+1

[Klaus Brockhaus]

In the OEIS there are a few sequences of the type
	Primes of form k*n+1
or equivalently
	Primes congruent to 1 (mod k)
for fixed positive integer k,
e.g. A002476 for k=3 and k=6, A002144 for k=4, A030430 for k=5 and k=10, A007519 for k=8, A061237 for k=9, A068228 for k=12.
In none of these entries the infiniteness of the sequence is mentioned as a problem, so I suppose that it is (a consequence of) an elementary or well-known theorem. Could someone kindly give a reference or a proof? And what is known about the density of such sequences relative to all primes (A000040)?

Thanks,
Klaus