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 $n \in \mathbb{Z}^{+}$.

Erdös and Kalmár found in 1939 a proof by induction 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.

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.

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 $n \in \mathbb{Z}^{+}$.

Erdös and Kalmár found in 1939 a proof by induction 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 $n>2$.

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 $n \in \mathbb{Z}^{+}$.

Erdös and Kalmár found in 1939 a proof by induction 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.