I have hard time proving obvious theories. For my Analysis class they asked me to proof that ever cut in R+ is a dedekind cut. I know these are the defintions: A Dedekind cut is a pair (A, B), where A and B are both subsets of rationals. This pair has to satisfy the following properties.
- A is nonempty.
- B is nonempty.
- If a ∈ A and c < a then c ∈ A.
- If b ∈ B and c > b then c ∈ B.
- If b not∈ B and a < b, then a ∈ A.
- If a not∈ A and b > a, then b ∈ B.
- For each a ∈ A there is some b > a so that b ∈ A.
- For each b ∈ B there is some a < b so that a ∈ B.
But how can I actually proof that?