How to find sup X and inf X

Question

How to find $\sup X$ and $\inf X$, if

$$X=\left\{ \frac{1}{n + m}: n, m\in \mathbb {N} \right\}.$$

Answer 1

Guide:

The smaller $n$ and $m$ are, the bigger $\frac{1}{n+m}$ is. What are the smallest value that $n$ and $m$ can take?

What happens as $n$ and $m$ are very huge?

Answer 2

Let $x = \frac{1}{n+m}$, then S :{$\frac{1}{n+m}$ , ..., $\frac{1}{3}$,$\frac{1} {2}$,$\frac{1}{1}$, $\frac{1}{n+m}$} $x$ $\in$ S, n,m $\in$N.

By definition, the supremum is upper bound of a for all $x$ $\in$ $S$. then $a$ $>=$ $x$ in S. the infimum is lower bound of a for all $x$ $\in$ $S$. then $b$ <= $x$ $\in$ $S$.