How to prove convergence to $1-\frac{1}{\log(n)}$

Question

I am trying to prove that:

$$\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\frac{1}{5}+\frac{4}{5}\left(...+\frac{p_{m-2}-1}{p_{m-2}}\left(\frac{1}{p_{m-1}}+\frac{p_{m-1}-1}{p_{m-1}}\left(\frac{1}{p_{m}}\right)\right)...\right)\right)\right)\right)\approx1-\frac{1}{\log(n)}$$

Where $m=\pi(\sqrt{n})$

Some help would be really welcomed! Thanks in advance.

Answer 1

I have found that Merten's third theorem, which remarkably can be proved without the assistance of the Prime Number Theorem, establishes that:

$\lim_{x\to \infty}\log{(x)}\prod_{p\le{x}}\left(1-\frac{1}{p}\right)=e^{-\gamma}$

Using this theorem, the proof I was looking for follows easily.