Lebl-Basic-Analysis-I · Problem 1.2.10

The key insight is that products of bounded nonnegative numbers stay bounded, and the supremum and infimum of products are determined by the suprema and infima of the factors. For the supremum, any element $ab \in C$ satisfies $ab \leq (\sup A)(\sup B)$ because each factor is bounded above by its supremum. For the infimum, use a similar argument with lower bounds—the smallest possible product occurs when both factors are at their minimum. The main work is verifying these bounds are actually achieved (or approached) and that no smaller bounds work.

The key insight is that products of bounded nonnegative numbers stay bounded, and the supremum and infimum of products are determined by the suprema and infima of the factors. For the supremum, any element $ab \in C$ satisfies $ab \leq (\sup A)(\sup B)$ because each factor is bounded above by its supremum. For the infimum, use a similar argument with lower bounds—the smallest possible product occurs when both factors are at their minimum. The main work is verifying these bounds are actually achieved (or approached) and that no smaller bounds work.