In the section 2. of this paper it is written that,
...The prime number theorem ensures that we can choose $B$ as close to $1$ as we want, provided $x_0$ is sufficiently large.
I think that there should be a subtle correction. The prime number theorem implies that there we can take both $A$ and $B$ arbitrarily close to $1$, not only $B$ as has been stated in the paper. Besides I don't think there is any PNT-independent method to prove the inequality $\dfrac{x}{\ln x}<\pi(x)$ for all sufficiently large $x$. Because as I have mentioned in this question that the paper doesn't seem to prove the desired inequality of the question without using PNT.
Am I wrong somewhere?
The Prime Number Theorem implies that both A and B can be taken arbitrarily close to 1; moreover, A can be taken as 1. This does not require a correction to the paper because its claims are correct.