Let $A=\{n+\frac{5}{n}:n\in \mathbb{N}\}$. In my lecture notes writes that $2\sqrt{5}$ is not a minorant of set A, so I'm interested to know if this is correct since by A-G inequality we have $n+\frac{5}{n}\geq 2\sqrt{5}$. It is obvious that $2\sqrt{5}$ is not infimum, but why it is not a minorant of this set?
Any help is welcome. Thanks in advance.
If $a,b>0$, you have $\frac{a+b}2=\sqrt{ab}$ if and only if $a=b$. So, since (if $n\in\Bbb N$) you never have $n=\frac5n$, $n+\frac5n>2\sqrt5$.