I've heard that there is an easy way to derive the asymptotic $\prod_{p\le x} \left(1-\frac{1}{p}\right) \sim \frac{c}{\log(x)}$ if one isn't interested in deriving $c=e^{-\gamma}$. I don't see how to do this, however. Does anyone here know where I could find a simple proof of this statement or even write down a proof for me?
I'm quite new to number theory, so if you only assumed minimal background, that would be very helpful. Thanks for your help!