Download I. Existence of Real Numbers

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.