Find supremum and infimum of $\{\arctan n\}^{\infty}_{n=1}$

Question

I want to find supremum and infimum of $\{\arctan n\}^{\infty}_{n=1}$ and decide if maximum and minimum exist. I think I have found supremum and infimum but how do I determine if max and min exist?

Since $\arctan$ is growing $n=1$ should be infimum and $n=\infty$ should be supremum, therefore I got $\sup=\frac{\pi}{2}$ and $\inf=\frac{\pi}{4}$. But is there a maximum and a minimum?

How I reason: If I remember correctly, the function $\arctan x$ does not have a maximum or minimum, but since we have the lowest value $n = 1$ here, infimum could also be a minimum? But since it is growing and moving towards infinity, it has no maximum.

Answer 1

Since $\arctan(a)+\arctan(1/a) =\pi/2 $ when $a > 0$, $\arctan(n) =\pi/2-\arctan(1/n) \to \pi/2 $ as $n \to \infty$ and $\arctan(n) \lt \pi/2 $ for all $n$.