# Thread: dedekind cuts and l.u.b. property

1. ## dedekind cuts and l.u.b. property

Here is the question:

Given $\displaystyle X=\{x_j\}_{j\in J}\in\Re$ (X is the set of all Dedekind cuts, sorry if that's obvious), show that the least upper bound of $\displaystyle X=\bigcup_{j\in J}x_j$

Is the least upper bound supposed to be a Dedekind cut itself, so of the form $\displaystyle x_i$, or is it an element $\displaystyle m\in\Re$? I think I understand how to do this otherwise, but any guidance would be helpful. Thank you!

2. I think that your posting shows a good deal of confusion.
I have taught courses in which the DEDEKIND CUTS have been central to the basic introduction to the structure of the real numbers. However, what you have posted has no relation to any notation about “CUTS” that I have ever seen.

Let me give you a standard (and elementary) reference for this topic.
Look in your mathematics library for PRICIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin.

3. The textbook we are using is called Chapter Zero: Fundamental Notions of Abstract Mathematics by Carol Schumacher, and that is the notation used for Dedekind cuts, in case you want to see that notation. I will definitely check out your recommendation! I think the point of this exercise is to show that the set of Dedekind cuts satisfies the l.u.b. property. Does that question make sense, at least? Thank you for your help!