Show that there exists $C>0$ such that the following holds
$$\Pi_{p \leq x} (1-\frac{1}{p})^{-1} = C \; \log(x) + O(1).$$ $\\$
I was thinking of writting $$\Pi_{p \leq x} (1-\frac{1}{p})^{-1} = \Pi_{p \leq x} (1+\frac{1}{p}) \Pi_{p \leq x} (1-\frac{1}{p^2})^{-1}$$ and $\Pi_{p \leq x} (1-\frac{1}{p^2})^{-1}$ clearly converges, wherease $\Pi_{p \leq x} (1+\frac{1}{p})=O(\log(x))$. But this doesn't yield the required result? (or maybe I don't understand the big-O notation correctly?) The second thought was to use a similar approach as to prove that $$\sum_{p \leq x} \frac{1}{n} = \log(x) + O(1)$$ but I don't know exactly how to translate the approach to my problem...
Help would be much appreciated :)