Euclid proof for infinitely many primes congruent to seven modulo eight

Suppose that $q_1,\dots ,q_r$ listed all prime numbers congruent to seven modulo eight. Put $m=q_1\cdots q_r$, with the empty product equal to one, and defineIf an odd prime $\ell$ divides $N$, then $(8m{)}^2\equiv 2(\mathrm{mod}\ell )$, so the second supplementary law for quadratic reciprocity forces $\ell \equiv 1$ or $7(\mathrm{mod}8)$. Since $N$ is seven modulo eight, at least one of its prime factors is seven modulo eight. But $N\equiv -1(\mathrm{mod}{q}_i)$ for every listed prime, so this prime factor is new. The contradiction proves infinitude.