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MATH 450/710: Worksheet Fall 2008 I. Existence of Real Numbers In this section, we assume the existence of the ordered field Q. To say that Q is a field means we have a set Q with operations + and · such that the pair (Q, +) forms an abelian group, whose identity is denoted by 0, and such that the pair (Q \ {0}, ·) is an abelian group, whose identity will be denoted by 1. These operations are related by the distributive laws: for all x, y, z ∈ Q (x + y) · z = x · y + y · z, and x · (y + z) = x · y + x · z. (1) In addition, there is a total ordering < of Q satisfying for all x, y, z ∈ Q • If x < y then x + z < y + z. • If x < y and z > 0 then x · z < y · z. The field Q together with the order relation < makes Q into an ordered field. This worksheet will guide you through the main steps needed to establish the existence of R as an ordered field with the least upper bound property. (Reference: §4 of Munkres.) 1. Construct the set R. 2. Define the operations + and · that make R into a field. 3. Define the relation < that makes (R, <) into an ordered field. 4. Prove that (R, <) has the least upper bound property. Let C be the collection of Cauchy sequences in Q; in other words, a sequence {xk }∞ k=1 of rational numbers belongs to C if and only if for every (rational) ε > 0 there exists an (integer) N such that for all k, ℓ > M we have |xk −xℓ | < ε. Let C0 be the collection of “null” sequences in Q, i.e. those sequences {xk }∞ k=1 of rational numbers such that for all ε > 0 there exists N such that |xk | < ε for all k > N. We have operations + and · on C and C0 defined by componentwise addition and multiplication. Scalar multiplication by a rational is also defined. 1a. Show that C0 ⊂ C. 1b. Show that x, y ∈ C and λ ∈ Q implies λx ∈ C and x+y ∈ C. (Conclude: C is a vector space over Q.) 1c. Repeat the previous exercise for C0 . 1d. For x, y ∈ C, define x ∼ y if x − y ∈ C0 . Show that this defines an equivalence relation on C. Let R be the set of equivalence classes for the relation ∼. (Algebraists would refer to R as the set of cosets of C0 .) Let C ∗ = C \ C0 . 2a. Show that for any x, y, x′ , y ′ ∈ C and for any λ ∈ Q if x ∼ x′ , y ∼ y ′ then λx ∼ λx′ and x + y ∼ x′ + y ′. (Conclude: there is a well-defined operation + on R that makes it into an abelian group. What is its identity element?) 2b. Show that every Cauchy sequence is bounded, i.e. for any sequence {xk }∞ k=1 in C there exists an M such that |xk | ≤ M for all k ≥ 1. 2c. Show that for any x, y, x′ , y ′ ∈ C such that x ∼ x′ and y ∼ y ′ we have x · y ∼ x′ · y ′. ∗ 2d. Show that for any sequence {xk }∞ k≥1 in C there exists an ε > 0 and an N such that for all k > N we have |xk | > ε. 2e. Show that for any x ∈ C ∗ there exists y ∈ C∗ such that x · y ∼ 1 where 1 denotes the constant sequence of ones. (Conclude: there is a welldefined operation · on R that, together with the operation + obtained in 2a., makes R into a field.) 3a. Show that for any x, y ∈ C that represent distinct elements of R, the sequence x − y is either eventually positive or eventually negative. 3b. Show that for any x, y, x′ , y ′ ∈ C such that x ∼ x′ and y ∼ y ′ we have x − y is eventually positive iff x′ − y ′ is eventually positive. 3c. Given x, y ∈ R define x < y if there are representatives x ∈ x and y ∈ y such that y − x is eventually positive. Show that the relation < on R is well-defined. 3d. Show that the < defines a total ordering on R. Let R be equipped with the order topology induced by <. Then we have a notion of convergence for sequences in R, namely, a sequence {xj } of elements in R converges to an element x ∈ R if for any open interval I ⊂ R that contains x there is an N such that for all j > N, xj ∈ I. 4a. Show that a bounded, monotone increasing sequence in Q is a Cauchy sequence. (Hint: use the Archmedian property of Q.) 4b. Let x = {xk } be a bounded, monotone increasing sequence in Q. By the previous exercise, x ∈ C and thus represents some element x ∈ R. For each k, let xk ∈ R be the element represented by the constant sequence whose terms are all equal to xk . (Note that a constant sequence of rationals is obviously Cauchy.) Show that the sequence {xk } converges to the element x in R, i.e. for any interval I = (a, b) that contains x there exists N such that for all k > N we have a < xk < b. (Conclude: every bounded, monotone sequence in Q converges in R.) 4c. Show that each element of R may be represented by a bounded, monotone increasing sequence in Q. 4d. Let {xj }∞ j≥1 be a monotone increasing sequence of elements in R. Represent each xj by a bounded, monotone increasing sequence in Q. Let xj,k denote the kth element of the sequence xj . Show that there exists an increasing sequence {Nj }∞ j≥1 of positive integers such that the ∞ sequence y = {yk }k≥1 of rationals defined by yk = xj,k for Nj−1 < k ≤ Nj , where xj,k denotes the kth element of the sequence xj and N0 = 0 by convention, is a bounded, monotone increasing sequence of rationals. 4e. Show that the sequence {xj } in the previous exercise converges to the element y of R represented by y. (Conclude: every bounded, monotone sequence in R converges in R.) 4f. Show that (R, <) has the least upper bound property. II. Existence of Rationals In this section, we assume a set N is given that satisfies the Peano’s axioms: (a) There is a distinguished element 1 ∈ N. (b) Successor function: There is a map σ : N → N that sends every n ∈ N to another element of N, called the successor of n. The map is injective and for every n ∈ N, σ(n) 6= 1. (c) Induction axiom: Suppose that S is a subset of N that has the properties: (i) 1 ∈ S, (ii) if n ∈ S then σ(n) ∈ S. Then S = N. For any n ∈ N, let the successor of n be denoted by n† . Let the operations + and · on N be defined recursively by: m + 1 = m† and m + n† = (m + n)† m · 1 = m and m · n† = m · n + m The following exercises, known as “Peano playing”, establishes the associative, commutative and distributive laws of natural numbers. 5a. Prove that (a + b) + n = a + (b + n) for any a, b, n ∈ N. (Hint: First prove it for n = 1 and all a, b; then induct on n.) 5b. Prove that a + b = b + a for all a, b ∈ N. 5c. Prove that m† · n = m · n + n for all m, n ∈ N. 5d. Prove that 1 · n = n for all n ∈ N. 5e. Prove that m · n = n · m for all m, n ∈ N. 5f. Prove (m · n) · p = m · (n · p) for all m, n, p ∈ N. 5g. Prove (a + b) · c = a · c + b · c. 6a. Let ∼ be the relation on N × N defined by (a, b) ∼ (c, d) whenever a + d = b + c. Show that ∼ is an equivalence relation on N × N. 6b. Let Z denote the set of equivalence classes of ∼. Show that if (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ) then (a+c, b+d) ∼ (a′ +c′ , b′ +d′ ). (Conclude: there is a well-defined operation + on Z.) 6c. Show that (Z, +) is an abelian group. 6d. Define an operation · on Z that makes it into a ring with unity. 6e. Let 0 denote the identity of (Z, +). Let Q = Z × (Z \ {0}). Define an equivalence relation on Q by (a, b) ∼ (c, d) whenever ad = bc. 6f. Let Q be the set of equivalence classes. Define an operation + on Q that makes it into an abelian group. 6g. Define an operation · on Q that together with + makes Q into a field.