Supremum and infimum of $\{\frac{m}{n}+\frac{1}{m}+\frac{1}{n}+\frac{n}{m}: m,n\in\mathbb{N}\}$

Question

Let $$S=\left\{\frac{m}{n}+\frac{1}{m}+\frac{1}{n}+\frac{n}{m}: m,n\in\mathbb{N}\right\}.$$ I think that $\sup S =\infty$ since the set is unbounded from above. By AG inequality, $\inf S =2$.

Am I right?

Any help is welcome.

Answer 1

Your answer is correct. However using the AM-GM Inequality, you have got a lower bound only. You cannot determine the $\operatorname{inf}$ using AM-GM Inequality

To determine the $\operatorname{inf}$ use the following.

When $m=n$, then $$\frac{m}{n}+\frac{1}{m}+\frac{1}{n}+\frac{n}{m}=2+\frac{2}{n}$$

Now use the fact that $$\lim_{n \to \infty} 2+\frac{2}{n}=2$$