Find the infimum and supremum of the set $S = \big\{ \tfrac1x - \tfrac1y:\, x,y \in \Bbb N \big\}$

Question

Find the infimum and supremum of the set $S = \big\{ \tfrac1x - \tfrac1y:\, x,y \in \Bbb N \big\}$.

My attempt: We know that the supremum means the smallest upper bound of the set. At $x=1$ and as $n$ goes to infinity it is obvious that the supremum is $1$.

Proof : Let $M=\sup S =1$. $$\frac1x - \frac1y \leq M$$

Answer 1

For a complete proof, guess the supremum and infimum either from intuition or calculations and prove two things:

(1): Name the supremum and infimum by $M$ and $m$, respectively. Prove that$$m\le {1\over x}-{1\over y}\le M$$ (2): find two sequences $a_{m,n}$ and $b_{m,n}$ such that$$a_{m,n}={1\over x_m}-{1\over y_n}\\b_{m,n}={1\over \hat x_m}-{1\over \hat y_n}\\a_{m,n}\to M\\b_{m,n}\to m$$

Answer 2

hint

$$-1\le \frac{-1}{y}<\frac 1x-\frac 1y<\frac 1x\le 1$$

$$\lim_{n\to+\infty}(\frac 11-\frac 1n)=1$$

and

$$\lim_{m\to+\infty}(\frac 1m-\frac 11)=-1$$ thus $$\sup S=1 \; \text{ and } \; \inf S=-1$$