Supremum of Suprema over Overlapping Domains

From ProofWiki
Jump to navigation Jump to search

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




Sources