On this site a proof that the density of primes is 0 is claimed. However, I am confused by multiple steps in this proof.
Firstly why does $\lim_{x\rightarrow\infty}\phi(x)/x=0$ imply the density is 0? Is this because $\phi(p)+1=p$ or because primes are co-prime to most numbers? I'm not sure if those are valid explanations.
Secondly, is the taking of the limit justified? As $x\rightarrow\infty$ there will still be primes not dividing $x$, infinitely many in fact.
Finally are we justified in taking $\zeta(1)^{-1}=0$? Is this shown from taking the limit of the product?
The last two points I'm mostly ok with but I don't see how they derive the first implication and so will accept any answer which explains this.