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.