Let $a$ and $q$ be two relatively prime positive integers. If I know that $$\lim_{s\rightarrow 1^{+}}\sum_{p\equiv a \bmod q}\frac{1}{p^s}=+\infty,$$ then it is clear that it tells me that there are infinitely many primes in the arithmetic progression $\{a+kq:k\in\mathbb{N}\}$. I wonder if there is an easy way to deduce that actually
$$\lim_{x\rightarrow+\infty}\sum_{\substack{p\leq x\\p\equiv a \bmod q}}\frac{1}{p}=+\infty.$$
My problem is that if I assume by contradiction that $\sum_{p\equiv a \bmod q}\frac{1}{p}$ converges, then nothing tells me that it should be equal to $\lim_{s\rightarrow 1^{+}}\sum_{p\equiv a \bmod q}\frac{1}{p^s}$, or at least nothing that I know. How can this be proven?