Determine whether the following subsets of $\mathbb{R}$ are bounded above or below. Moreover find the Supremum and Infimum (if they exist) and decide whether it is a maximum or a minimum.
Is $A=\{2^{-n}+1/m: n,m\in \mathbb{N}\}$ bounded?
Sadly, I don't know how to start with these problems. Can you please explain?
Why not?
Just observe $(\forall m):\;\frac{1}{m} \leq 1$ and $(\forall n): \frac{1}{2^n} \leq \frac{1}{2}$. Use these two to conclude the bounded ness
To make the problem easier, write your set as $C+B$ where $C=\{\frac{1}{2^n}: n \in \Bbb N \}$ and $B=\{\frac{1}{m}: m \in \Bbb N \}$ and use $\sup (C+B)=\sup C +\sup B$ if they exist!
For infimum, use $\inf (C+B)= \inf C+ \inf B$.