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

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.

• Yes, this is correct. The sequence is strictly increasing, so it clearly has no maximum, and the first element is a minimum.
– Mark
Sep 2, 2021 at 23:08

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 $$n$$.