= Euler proof that the sum of reciprocals of primes diverges
{c}
For a finite set of primes,
$$
\prod_{p\leq x}\left(1-\frac1p\right)^{-1}
$$
expands as the sum of $1/n$ over positive integers whose prime factors are at most $x$. It contains every term with $n\leq x$, so these partial products dominate the <harmonic series> and diverge. Since
$$
-\log(1-t)=t+O(t^2)
$$
uniformly for $0\leq t\leq1/2$, convergence of $\sum_p1/p$ would force convergence of the logarithms of these products, a contradiction.
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