TITLE: What was the first elementary proof that $\pi(x)=o(x)$?
QUESTION [5 upvotes]: 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 $\pi(x)=O(x/\log x)$.

REPLY [4 votes]: 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.