Prime Number Theorem in Eulerian Logarithmic Integral Form

Theorem
The Prime Number Theorem is equivalent to:
 * $\ds \lim_{x \mathop \to \infty} \frac {\map \pi x} {\map \Li x} = 1$

where:
 * $\map \pi x$ is the prime-counting function
 * $\map \Li x$ is the Eulerian logarithmic integral:
 * $\ds \map \Li x := \int_2^x \dfrac {\d t} {\ln t}$

Proof
Using Integration by Parts:

We have that $\dfrac 1 {\paren {\ln t}^2}$ is positive and decreasing for $t > 1$.

Let $x \ge 4$.

Then:

Then:

Then: