Chebyshev's bounds

Bounds. There exist constants $0<a<1<b$ such that for $x$ large enough:
$$a\frac{x}{\ln x}\le\pi(x)\le b\frac{x}{\ln x}.$$

Apply at $x=1000$. $1000/\ln1000\approx145$. So $\pi(1000)$ is within a constant factor of $145$; actual value is $168$.

Conclusion. Chebyshev's bounds prove that $\pi(x)$ grows like $x/\ln x$ up to a constant, a first step toward the prime number theorem.