1.

Consider the set S=Q, the rational numbers.

a. Give a cut SU, SL of S such that SU has a minimum but SL has no maximum.

b. Give a cut SU, SL of S such that SL has a maximum but SU has no minimum.

2.

Prove that N is order complete by following the outline here. Provide the justification for each step.

1. Suppose SL and SU make a cut of N.

2. If n SL, then so does every m N satisfying m < n.(Hint: suppose not, then since SLSU=N, then the omitted m must belong to SU; use this to derive a contradiction against SL, SU being a cut.)

3. If n U, then so does every m M satisfying m > n.(Hint: similar to the previous step.)

4.SL is finite.

5. SL has a maximum.

6. max SL+ 1 is in SU, and is its minimum