Prove indexed set and Cartesian product

Question

I have a problem as follows:

And this is my work:

Can some one suggest a way to do the opposite direction:

Starting with: Conversely, suppose $P\ \in\bigcup{\ (A}_i\times\ S)$ for some $i\ \in I$.

Thank you so much!

Answer 1

Suppose $P \in \bigcup\limits_{i \in I} (A_i \times S)$. Then take some $i \in I$ such that $P \in A_i \times S$. Then we can write $P = (a, s)$ for $a \in A_i$ and $s \in S$. Then $a \in \bigcup\limits_{i \in I} A_i$. Then $P = (a, s) \in (\bigcup\limits_{i \in I} A_i) \times S$.

Thus, we see that $\bigcup\limits_{i \in I} (A_i \times S) \subseteq (\bigcup\limits_{i \in I} A_i) \times S$.