Suppose an equivalence relation is defined as: for $a, b \in \mathbb{R}$, $a R b \iff a - b = n/m$, where we have $n, 0 \neq m \in \mathbb{Z}$. For any natural number $N$ and any $a\in \mathbb{R}$, show that there is a $b\in \mathbb{R}$ so that $b R a$ and $b< \frac{1}{N}$ and $b \geq 0$.
I seem to struggle with "existence" proofs and I'm new to proofs using analysis. For the second part, I think I'm supposed to use the archimedian property/ies but I'm not very sure where $0$ and $\frac{1}{N}$ come from. I'd appreciate any hints or assistance!
Intuitively, you are being asked to show that if you are given any real $a$ you can find a rational $\frac nm$ such that $0 \le a-\frac nm \lt \frac 1N$ or in other words you can find a rational as close as needed to any real. Note that if $m$ is huge the spacing between the options is very small. In fact, if $m \gt N \ldots$