I'd like to be able to find $\inf A$ and $\sup A$ for the set $A = \{\cos{(\frac{\pi}{2}\lceil\root\of{n}\rceil)} - \cos{(\frac{\pi}{2}\root\of{n})} : n \in N_+\}$
I think that I know intuitively that the infimum and supremum will be $-1$ and $1$ respectively as it is clear to me from the ceiling function that the maximum - but unobtainable - difference in multiples of $\frac{\pi}{2}$ for the elements of $A$ is $1$, and investigating the graph of cosine would suggest that if we pick any two points on it no more than $\frac{\pi}{2}$ apart, we cannot exceed the infimum or supremum I have said.
Does anyone know how to prove this formally, and preferably in not too complex a way? Thank you!