1.

Prove that the set{1,1,1/2,1/2,1/3,1/3,1/4,1/4, . . .}is not order complete.

(First ask yourself, where should you cut?)

2

.

Prove that Nis order complete by following the outline here. Provide the justification

for each step.

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

b. 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.)

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

d. SL is finite.

e. SL has a maximum.

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