Sum of Indexed Suprema

Theorem

Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the real numbers $\R$ indexed by $J$.

Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$.

Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively.

Then:

$\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) + \left({\sup_{S \left({j}\right)} b_j}\right) = \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i + b_j}\right)}\right)$

Proof

 $\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) + \left({\sup_{S \left({j}\right)} b_j}\right)$ $=$ $\displaystyle \sup_{R \left({i}\right)} \left({a_i + \sup_{S \left({j}\right)} b_j}\right)$ Sum with Maximum is Maximum of Sum $\displaystyle$ $=$ $\displaystyle \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i + b_j}\right)}\right)$ Sum with Maximum is Maximum of Sum

$\blacksquare$