Prove that ⋃ n ∈ N ( − n , 1 n ] = (...
I understand that for this to be true, I have to show that and , however, I'm stuck on figuring it out. I think the latter can be proven by contradiction somehow.
Answer & Explanation
Show mutual inclusion.
First suppose that . Choose such that . Since , . This shows that .
Conversely, suppose that . If , we may choose such that . Then , so . On the other hand, if , there exists such that . , so , so . This shows that .
as and as then
For the second, we know that for every number a there exists number such that , in the same way it works backwards, so that for every a there exists such that .
If we take , else we take where y is a smaller number than x that is in and satisfies that , so we get that again .