Product of Indexed Suprema of Non-Negative Numbers

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 $\family {b_j}_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$.

Let $\map R i$ and $\map S j$ be propositional functions of $i \in I$, $j \in J$.

Let $\ds \sup_{\map R i} a_i$ and $\ds \sup_{\map S j} b_j$ be the indexed suprema on $\family {a_i}$ and $\family {b_j}$ respectively.

Then:

$\ds \paren {\sup_{\map R i} a_i} \paren {\sup_{\map S j} b_j} = \sup_{\map R i} \paren {\sup_{\map S j} \paren {a_i b_j} }$

$\ds \paren {\sup_{\map R i} a_i} \paren {\sup_{\map S j} b_j}$

$=$

$\ds \sup_{\map R i} \paren {a_i \times \sup_{\map S j} b_j}$

$\ds $

$=$

$\ds \sup_{\map R i} \paren {\sup_{\map S j} \paren {a_i b_j} }$

$\blacksquare$