# 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

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The results upon which this depends need to be enhanced to encompass the full supremum, not just the maxYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

\(\ds \paren {\sup_{\map R i} a_i} + \paren {\sup_{\map S j} b_j}\) | \(=\) | \(\ds \sup_{\map R i} \paren {a_i + \sup_{\map S j} b_j}\) | 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

- 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 $35$