Lebl-Basic-Analysis-I · Problem 1.2.9

The key insight is that suprema and infima are additive when you're combining elements from independent sets. To show $C$ is bounded, use the fact that $A$ and $B$ are bounded to establish upper and lower bounds on all possible sums. For the supremum formula, prove two things: first, that $\sup A + \sup B$ is an upper bound for $C$ (by the definition of supremum), and second, that no smaller number works as an upper bound (by the minimality property of supremum). The infimum follows by the same logic applied to lower bounds.

The key insight is that suprema and infima are additive when you're combining elements from independent sets. To show $C$ is bounded, use the fact that $A$ and $B$ are bounded to establish upper and lower bounds on all possible sums. For the supremum formula, prove two things: first, that $\sup A + \sup B$ is an upper bound for $C$ (by the definition of supremum), and second, that no smaller number works as an upper bound (by the minimality property of supremum). The infimum follows by the same logic applied to lower bounds.