infinitely many primes of the form 6k + 5 and 6K + 1

hey the question is

show that there are infinitely many primes of the form 6k + 5. does the method work for 6k + 1.

my answer so far is

suppose there are finite primes of the form 6k + 5

order them such: p(1) < p(2) <....< p(n)

let R = 6(p(1)p(2)...p(n)) + 5

R can't be prime, if it is R > p(n)

R can't be composite as any division will give a remainder of 5

therefore there are infinitely many primes :)

i think there's something not quite right with it and i can't use Dirichlet's Theorem

kudos for any help! :)