Show that, for $x \geqslant 2$ a real number,

$$\prod_{\substack{p \leqslant x \\ p \text { is prime }}}\left(1-\frac{1}{p}\right)^{-1}>\log x$$

Hence prove that

$$\sum_{\substack{p \leqslant x, p \text { is prime }}} \frac{1}{p}>\log \log x+c,$$

where $c$ is a constant you should make explicit.