Can we prove that for some $k$ we have the following inequality?

Question

I want to prove that that for $k > 3, k \in \mathbb{N}$ there is some $p_k$ which is a prime number and we have that $k < p_k \leq 2k-3$. Now to show this I want to use Bertrand's postulate, but I am not sure how. I have seen the proof in the book "PROOFS FROM THE BOOK", and the proof is quite involved, however I am thinking that the 5th step in this proof should be helpful (maybe some derivative estimation). Any help is appreciated

Answer 1

This is already Bertrand's postulate, see wikipedia for a reference with proof:

Bertrand's postulate: For any integer $n>3$, there always exists at least one prime number $p$ with $$ {\displaystyle n<p<2n-2.} $$ There are several stronger statements known, e.g., by Dusard, $$\frac{x}{\ln x}\left(1+\frac{0.992}{\ln x}\right) < \pi(x) < \frac{x}{\ln x}\left(1+\frac{1.2762}{\ln x}\right)$$ for all $x\ge 599$. This implies Bertrand's postulate above. For a reference, see here:

Weak versions of Bertrand's postulate

Also on MSE there are some links. This is sometimes called the "stronger form", see here.