1549901369-Elements_of_Real_Analysis__Denlinger_

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30 Chapter 1 • The Real Number System


In this sense, every ordered field contains the familiar natural numbers,
integers, and rational numbers.


EXERCISE SET 1.4

In each of the following, assume that F is an ordered field. Prove the stated
property.



  1. Z satisfies all the properties of an ordered field except (M4). [You may
    assume (AO) and (MO); see Exercise 13 below.]

  2. Ql is the "smallest" ordered field, in the sense that every ordered field F
    contains Ql as a subset.

  3. Let n E Z. If n^2 is divisible by 2, then n is also divisible by 2. [Hint: any
    integer is either even (of the form 2k, where k E Z) or odd (of the form
    2k + 1, where k E Z).]

  4. There is no element of Ql whose square is 3. [Hint: Use suitable modifi-
    cations of Exercise 3 and Theorem 1.4.5.]
    In Exercises 5- 10 , assume that F is an ordered field with at least one
    irrational element.

  5. If x is rational and y is irrational, then x + y is irrational.

  6. If x =I- 0 is rational and y is irrational, then xy is irrational.

  7. If x is irrational, then so are -x and x-^1.

  8. The set of irrationals is not closed under addition.

  9. The set of irrationals is not closed under multiplication.

  10. F contains infinitely many irrational elements.

  11. The Principle of Mathematical Induction, and its alternative forms, can
    use any integer no as the "starting point." (Theorem 1.3.11 establishes
    this for natural numbers only.)

  12. Prove that Z satisfies axioms (AO) and (MO).

  13. (Project) Integer exponents: In Definition 1.3.12 we used mathemat-
    ical induction to define an, \:/a in an ordered field F and all n E N. For
    a =I- 0 we define a^0 = 1 and a-n = l/an. Prove that with this definition
    the familiar "laws of exponents" hold: if a, b are nonzero elements of an
    ordered field F, then \:/m, n E Z,
    (a) aman = am+n (b) (amt= amn
    (c) am /an= am-n (d) (ab)n = anbn
    [See Exercises 1.3.20- 1.3.22.]

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