  We prove that if $A$ is any set of prime numbers satisfying \[ \sum_{a\in
A}\frac{1}{a}=\infty, \] then $A$ must contain a $3$-term arithmetic
progression. This is accomplished by combining the transference principle with
a density increment argument, exploiting the structure of the primes to obtain
a large density increase at each step of the iteration. The argument shows that
for any $B>0$, and $N>N_{0}(B)$, if $A$ is a subset of primes contained in
$\{1,\dots,N\}$ with relative density $\alpha(N)=(|A|\log N)/N$ at least \[
\alpha(N)\gg_{B}\left(\log\log N\right)^{-B} \] then $A$ contains a $3$-term
arithmetic progression.
