Euler proof that the sum of reciprocals of primes diverges

For a finite set of primes,expands as the sum of $1/n$ over positive integers whose prime factors are at most $x$. It contains every term with $n\le x$, so these partial products dominate the harmonic series and diverge. Sinceuniformly for $0\le t\le 1/2$, convergence of ${\sum }_{p}1/p$ would force convergence of the logarithms of these products, a contradiction.