For an analytic number theory homework assignment, we are asked to prove the following (using the Prime number theorem $\pi(x) \sim x/\log(x)$ as $x \to \infty$ ):
For every $\epsilon > 0 $ and every positive integer $k$, there is a real $x_{0} (k, \epsilon)$ such that for every $x \geq x_{0} (k, \epsilon)$ there are at least $k$ prime numbers in the interval $[x, (1 + \epsilon)x]$.
I have no idea how to prove this statement. Do you have any hints or suggestions by means of which I can get started?
From PNT you have$$\pi\left(x\left(1+\epsilon\right)\right)-\pi\left(x\right)\sim\frac{x\left(1+\epsilon\right)}{\log\left(x\right)+\log\left(1+\epsilon\right)}-\frac{x}{\log\left(x\right)}\sim\frac{x\left(1+\epsilon\right)-x}{\log\left(x\right)}=\frac{x\epsilon}{\log\left(x\right)}\geq1$$ if $x$ is large enough.