Analysis. Supremum and infimum.

Question

Let $A,B \subset\mathbb{R}$ and $C =\{x+y | x ∈ A,y ∈ B\}$. How are the numbers $\inf A$, $\inf B$, and $\inf C$ related? How are the numbers $\sup A$, $\sup B$, and $\sup C$ related?

Answer 1

From the properties of the infimum,

$$ \inf(C) = \inf(A) + \inf (B) $$

To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.

Same applies for the supremum.