Prime Number Theorem in Eulerian Logarithmic Integral Form

Theorem
The Prime Number Theorem is equivalent to:
 * $\displaystyle \lim_{x \mathop \to \infty} \frac {\pi \left({x}\right)} {\operatorname {Li} \left({x}\right)} = 1$

where:
 * $\pi \left({x}\right)$ is the prime-counting function
 * $\operatorname {Li} \left({x}\right)$ is the Eulerian logarithmic integral:
 * $\displaystyle \operatorname {Li} \left({x}\right) := \int_2^x \dfrac {\mathrm d t} {\ln t}$

Proof
Using Integration by Parts:

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

Let $x \ge 4$.

Then:

Then:

Then: