The following reference will meet your purpose:
P. Dusart, The $k$th prime is greater than $k(\log k+\log\log k−1)$ for $k$ > 2, Math. Comp. 68 (1999), 411–415.
At the end of this paper, the author wrote that "for $x\ge3275$ the interval $[x,x+x/(2\ln^2x)]$ contains at least one prime". This implies your desired result.