1549901369-Elements_of_Real_Analysis__Denlinger_

Appendix C

Answers & Hints for

Selected Exercises

Caveat: The answers and hints provided here are intentionally brief. Complete
solutions and proofs will require more fully developed explanations. Individ-
ual instructors will have their own standards of completeness and will make
them known to their students. Most solutions given here will need considerable
amplification to meet these standards.
Some abbreviations used here are Thm. for Theorem, Cor. for Corollary,
Defn. for Definition, and Ex. for Example.

Chapter 1

EXERCISE SET 1.1-A

1. AO, Al, A2, MO, Ml, M2, M3, D.


2. All field axioms.


3. AO, Al, A2, A3, A4, but not MO. Ml- M4 and Dare not relevant.


4. AO, Al, A2, A3, A4, MO, D.


5. All field axioms.

EXERCISE SET 1.1-B

1. x =/= 0 :=> 3u E F 3 xu = ux = 1. Then xy = xz ==> u(xy) = u(xz) ==>
   (ux)y = (ux)z ==> ly = lz ==> y = z.


2. Suppose x =/= 0 and u, u' both have property described in (M4); i.e., xu = 1,
   and xu' = 1. Then u = ul = u(xu') = (ux)u' = lu' = u'.


3. Suppose x =!= 0. Then xx-^1 = 1 =!= 0, so by (d), x-^1 =!= 0.

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