Change of Index Variable of Supremum

Theorem

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.

Let $R \left({i}\right)$ be a propositional functions of $i \in I$.

Let $\displaystyle \sup_{R \left({i}\right)} a_i$ be the indexed supremum on $\left \langle {a_i} \right \rangle$.

Then:

$\displaystyle \sup_{R \left({i}\right)} a_i = \sup_{R \left({j}\right)} a_j$

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 $35$