1549901369-Elements_of_Real_Analysis__Denlinger_

1.2 The Order Properties 15

( c) Suppose F is an ordered field. Then 1 E F and, since F is closed under

addition, F must contain the elements

1, 1+1, 1+1+1, 1+1 + 1 + 1,

By Corollary 1.2.7, 1 E P , and by Axiom (01), each element in this list is

in P. Moreover, by Theorem 1.2.8,

1 < (1+1) < (1+1+1) < (1+1+1+1) <

By the transitive property, each successive element in the above list is larger
than all previous elements in the list. So, the list above contains no duplicates;
all elements are different.
Since there is no end to the number of times we can add 1, this list must
contain an infinite number of different elements. Therefore, P must be an infi-
nite set. •

EXERCISE SET 1.2-A

l. Which of the fields found in Exercise Set 1.1-A are ordered fields with

respect to some natural choice of subsets P. [In each case, try to find a

natural set P of "positive" elements.]

2. Prove Theorem 1.2.5 (a).


3. Prove Theorem 1.2.5 (b).


4. Prove Theorem l. 2. 5 ( c). [Hint: apply Axiom (03).]


5. Prove Theorem 1.2.6 (b).


6. Prove Theorem 1.2.6 (d).


7. Prove Theorem 1.2.6 (e)


8. Prove Theorem 1.2.6 (f).


9. Prove Corollary 1.2.7.


10. Prove Theorem 1.2.8 (b).


11. Prove Theorem 1.2.8 (d).

12. Prove the "~" part of Theorem 1.2.8 (e).

13. Prove Corollary 1.2.9.

14. Prove the '«~=" part of Theorem 1.2.10 (a).

15. Prove Theorem 1.2.10 (b).