Change of Index Variable of Supremum

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 $\map R i$ be a propositional functions of $i \in I$.

Let $\ds \sup_{\map R i} a_i$ be the indexed supremum on $\family {a_i}$.

Then:

$\ds \sup_{\map R i} a_i = \sup_{\map R j} a_j$

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 $35$