So I was reading a proof of Brun's theorem in AoPS, then I came to these two inequalities that I can not figure out how they have been obtained.
It states that if $p$ stands for prime numbers in the expressions below and $y\in\mathbb{N}$ then there exists a contant $C\in\mathbb{R}$ such that $$\prod_{p\le y}(1-\frac{1}{p})\le \frac{C}{(\ln y)^2}$$ Also $$\sum_{p\le y}\frac{1}{p}\le e\ln\ln y$$ The only fact I thought might be helpful to prove them was Prime number theorem but I was not able to find the way out. Any help would be appreciated!