Is there a prime between $k$ and $\dfrac{11}{9}k$, $\forall k\ge 24$?

Question

Given $k\in\mathbb{N}$, $k\ge 24$, is there always a prime number in the interval $\left[k,\dfrac{11}{9}k\right]$? I tried to verify this statement with the computer and it seems to hold. Is it possible to prove it?

Answer 1

There are various results the give absolute upper bounds on $\frac{p_{n+1}}{p_n}$, converging to $1$ and holding for sufficiently large $n$. See, for example, those by Dussart mentioned at Prime number theorem and Bertrand's postulate at Wikipedia.

Thanks to these results, every true statement of the general form you give is provable -- the general result gives an $n$ beyond which it is guaranteed to be true, and if your limit is smaller than that, there's a finite number of cases to check.