Math 310-01/02

11-3-2017

1. Prove that

I will show that each of the two sets is contained in the other.

Let . By definition of the union, for some . Then

Therefore, . This proves that .

Conversely, suppose , so

Note that . Hence, for some ,

For if, on the contrary, for all , then .

Therefore, for some . Hence, , by definition of the union. This proves that .

Hence, .

*All action is involved in imperfection, like fire in smoke.*
- *The Bhagavad Gita*

Copyright 2017 by Bruce Ikenaga