Generally most of the proof I found online are based on The Analytic Theorem , that proof was sketched by Von Newman. In a book I found a different proof.
The proof first derived $\int_2^x\frac{\psi(u)}{ u^2}du= \log x+O(1)$, then it showed- $$\varliminf \frac {\psi(x)}{ x}\leq 1 $$ $$\varlimsup \frac {\psi(x)}{ x}\geq 1 $$
Then the author used the fact- $$\pi(x)\sim \frac{\theta(x)}{\log x}\sim \frac{\psi(x)}{\log x}$$ Showing- $$\varliminf \frac {\pi(x)}{ x/\log x}\leq 1 $$ $$\varlimsup \frac {\psi(x)}{ x/\log x}\geq 1 $$ Saying that if $\frac {\pi(x)}{ x/\log x}$ tend to a limit as $x \to \infty$, then the limit must be 1, thus, $$\lim_{x \to \infty}\frac {\pi(x)}{ x/\log x}=1$$
what is the original source of this proof? Who gave above proof? Plz, give the reference (book, journal paper).