I assume you are all familiar with the idea of a "cut": A non empty subset A of with the following properties:
(1) A is bounded above
(2) A has no maximum (least upper bound)
(3) A is closed downwards: And
I'm not sure with the following things:
(a) Prove that every cut is "dense" (Jason: just comes out of the closed downwards property?)
(b) Prove that the union of two cuts is always a cut.
(c) Let be any collection of cuts. Prove that their union
is either a cut or else is equal to .
(d) [Define what it means for a linearly ordered set to be "complete". (I know this, if every subset of X which is bounded above has the least upper bound, X is complete) ]
Recall that the set of real numbers is defined as the set of all cuts. Prove that is complete.
(Jason: I know I have to use the previous part some how. Every subset of is a set of cuts and these are all bounded above, so is the proof just that these cuts have the least upper bound in ? And if so, how are they cuts?!)
(e) For a rational number r with the following definition
I need to prove that is a cut.
(f) Let denote the set of all positive rational numbers. Describe the intersection:
is it a cut? (Jason: I thought this might be something similar to the empty set? In which case it's not a cut.)
(g) Prove that every set that has a lower bound, also has a greatest lower bound. (Jason: here I think because is complete and then take the negative of all cuts?)
Help on any of this is much appreciated.