I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this.
Claim: Given any rational number r and any positive integer $n$, there exists a rational number on either side of $r$ and differing from $r$ by less than $\frac1n$.
My Proof so far: Any rational number can be written as the ratio of two integers. Let $r = \frac pq$ According to the claim, there exists another rational number, which we will call $r'$ that differs from $r$ by less than $\frac 1n$ Therefore, this can be written as: $|r'- r| < \frac 1n$
And this is where I had become stuck at.