Denote by $\pi(x)$ the number of primes $\leq x$. I'm interested in knowing who came up with the **first elementary proof** that $\pi(x)=o(x)$. 


I know that Chebyshev demonstrated elementarily before Hadamard and de la Vallee-Poussin the slightly stronger result that $\pi(x)=O(x/\log x)$.



What was the first elementary proof that $\pi(x)=o(x)$?
Leonhard Euler knew that the infinite product:


$$ \prod\_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right)^{-1} = \sum\_{n=1}^{\infty} \frac{1}{n} $$


is divergent (and used this to prove the infinitude of primes), so would have also known that the product:


$$ \prod\_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right) $$


tends to zero. In other words:


$$ \lim\_{n \rightarrow \infty} \prod\_{p < n\textrm{ prime}} \left(1 - \frac{1}{p} \right) = 0 $$


Observe that the product in the last expression is the density of integers which are coprime to the set $\{ p < n \textrm{ such that } p \textrm{ is prime} \}$; this trivially implies that primes have zero upper density.


