Supremum of Suprema over Overlapping Domains
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Theorem
Let $\family {a_i}_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.
Let $\map R i$ and $\map S i$ be propositional functions of $i \in I$.
Let $\ds \sup_{\map R i} a_i$ and $\ds \sup_{\map S i} a_i$ be the indexed suprema on $\family {a_i}$ over $\map R i$ and $\map S i$ respectively.
Then:
- $\ds \map \sup {\sup_{\map R i} a_i, \sup_{\map S i} a_i} = \sup_{\map R i \mathop \lor \map S i} a_i$
Proof
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Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $35$