TITLE: What is the simplest proof that the density of primes goes to zero?
QUESTION [51 upvotes]: By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$.
I want something much weaker, namely that the proportion just goes to zero, at whatever rate. And I want the easiest proof possible.
The simplest proof I know uses estimates involving the binomial coefficient $\binom{2n}{n}$, but the argument still feels a bit involved.
Does anyone know an even simpler proof that the density of primes goes to zero?

REPLY [6 votes]: In my opinion, the simplest way to establish that $$\lim_{n \to +\infty} \frac{\pi(n)}{n}=0$$ is via the elementary inequality
$$ \prod_{p \leq n} p \leq 4^{n-1} \qquad \mbox{(*)}$$
which holds for every $n \in \mathbb{Z}^{+}$.
In 1939, Erdös and Kalmár found a proof of this inequality "which comes out of THE BOOK" (cf. P. Erdös, Ramanujan and I. In: K. Alladi (ed.), Number Theory - Madras 1987. Lecture Notes in Mathematics, vol. 1395. Springer Verlag, p. 2.). Clearly enough, as an immediate consequence of the Erdös-Kalmár ineq. we have that
$$ \pi(n) < (2\log 4+1)\frac{n}{\log n}$$
for every integer $n>2$ and whence the result on the natural density of the primes.