After proving their existence, find inf A and sup A of the folllowing set
A={ x∈R|x=$(−1/2)^m\ $− $(3/n)$ for some m,n ∈ N\{$0$}} .
I' m trying to use the idea that sup(A+B) = sup A + sup B and same for inf (A+B) = inf A + inf B , so let
A = {(−1/2)^m\ for some m ∈ N\{$0$}} and B= { $(-3/n)$ for some n ∈ N\{$0$}}
After proving that A is a nonempty bounded set , i noticed that the sup is $1/4$ and that the inf is $-1/2$ but i'm having a hard time to prove that these are respectively upper and lower bounds of A.
Is it right if we write for any $m$ greater or equal to $1$ we've $(-1/2)^m$ is greater or equal to $-1/2$ ?
Any hints for showing that $1/4$ is an upper bound of A ?
I would suggest breaking up A into considering when $m$ is odd verses even. It's easy to determine the behavior in both of those cases, and the rest of the proof is easy.