TITLE: Prime number between $\sqrt{n}-n^{1/3}$ and $\sqrt{n}$
QUESTION [10 upvotes]: Can anyone give me a proof or a reference for a proof that there exists a prime number between $\sqrt{n}-n^{1/3}$ and $\sqrt{n}$, for $n$ sufficiently large? 
I am reading a lecture where the professor uses this fact but does not provide any reference.
Thank you!

REPLY [2 votes]: Even a stronger bound is proved by R. C. Baker, G. Harman, and J. Pintz in a paper “The difference between consecutive primes II.”. For all $x > x_0$ the interval $[x-x^{0.525},x]$ contains prime numbers. With enough effort, the value of $x_0$ could be determined effectively.