Let $\displaystyle \langle S _{i} | i \in I \rangle $ be an indexed system of sets; show that $\displaystyle \prod _{i \in I} S _{i} $ exists. [Hint: $\displaystyle \prod _{i \in I} S _{i} \subseteq P(I \times \bigcup _{i \in I} S _{i}) $ ]