TITLE: How to find sup and inf of this set?
QUESTION [0 upvotes]: I have this set $$X=\{\frac{\cos(\pi m)}{m}+\exp(-|n|),\, n\in \mathbb{Z}, m\in \mathbb{Z}^*\}$$
I know that $X$ is bounded, because for any m and n we have
$$
|\frac{\cos(\pi m)}{m}+\exp(-|n|)|\leq |\frac1m|+1\leq 2
$$
but how to find inf and sup of X?
Thank you

REPLY [0 votes]: So you know that $2$ is an upper bound for $X$.  Can you find a pair of values $(m,n)$ such that $$f(m,n) = \frac{\cos \pi m}{m} + e^{-|n|}$$ attains this upper bound?  If so, you have shown $\sup X = 2$, since the upper bound is tight.  If not, then what choice of $(m,n)$ gets you the largest possible value?
As for the infimum, can $f(m,n) < 0$?  What is $\inf e^{-|n|}$ in particular?  Then, for what choice of $m$ is $(\cos \pi m)/m$ as small as possible?
{"set_name": "stack_exchange", "score": 0, "question_id": 3858441}