Sum of Indexed Suprema

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Theorem

Let $\sequence {a_i}_{i \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\sequence {b_j}_{j \mathop \in J}$ be a family of elements of the real numbers $\R$ 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 $\sequence {a_i}$ and $\sequence {b_j}$ respectively.


Then:

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


Proof

\(\ds \paren {\sup_{\map R i} a_i} + \paren {\sup_{\map S j} b_j}\) \(=\) \(\ds \sup_{\map R i} \paren {a_i + \sup_{\map S j} b_j}\) Sum with Maximum is Maximum of Sum
\(\ds \) \(=\) \(\ds \sup_{\map R i} \paren {\sup_{\map S j} \paren {a_i + b_j} }\) Sum with Maximum is Maximum of Sum

$\blacksquare$


Sources