Find the supremum (if exists) of the set $\frac2{\sqrt n}+4$, $n\in\Bbb N$

Question

I already found the $\inf$ of the set but I dont know what about the $\sup$. Thanks in advance

Answer 1

6 is the ans as the sequence is clearly decreasing

Since

$2/\sqrt{n}+4>2/\sqrt{n+1}+4$ as

$\sqrt{n+1}>\sqrt{n}$

So $f(n)>f(n+1)$

Hence $f(1)>f(2)>f(3) \dots $

So $f(1)$ is the maximum, hence the answer!