Lecture 2

Econ 2001

2015 August 11

Lecture 2 Outline

1 Fields2 Vector Spaces3 Real Numbers4 Sup and Inf, Max and Min5 Intermediate Value Theorem

Announcements:- Friday’s exam will be at 3pm, in WWPH 4716; recitation will be at 1pm. Theexam will last an hour.

Number Concept

“A satisfactory discussion of the main concepts of analysis (such asconvergence, continuity, differentiation, and integration) must be based on anaccurately defined number concept.” (Rudin)

What does this mean? All those concepts use some idea of distance betweenpoints.

Distance is reasonably easy to think about when dealing with points on thereal line... but we want a concept that works for abstract sets.

So, we need a tool that can give meaning to the idea that points are “close”or “far”.

Concepts like close and far imply the existence of some well behaved unit ofmeasure (Rudin’s “number concept”).

This unit of measureis the set of real numbers.

Next, describe the properties one wants from an abstract number system.

Ordered SetsDefinitionLet A be a set. An order on A is a relation denoted by < with the following twoproperties

If x ∈ A and y ∈ A then one and only one of the following statements is true

x < y , x = y , x > y

If x , y , z ∈ Ax < y and y < z ⇒ x < z

An ordered set is a set on which an order is defined.

RemarkThese definitions make no mention of numbers: very abstract sets can havean order.

NotationThe notation x ≤ y indicates x < y or x = y or both.

Therefore x ≤ y means ¬ (x > y )

Fields

A number concept should allow basic operations like addition andmultiplication.

A binary operation combines two elements of a set to produce anotherelement of that set.

A binary operation on X is a function from X × X to X .

Spaces on which these operations are possible have special structure, so thatsuch functions are well defined, and have familiar and useful properties.

A field is a set in which addition and multiplication satisfy standardproperties, like commutative and associative, and where zero and one can bedefined and have the usual meaning.

elements of a field are “number-like”

R (the set of all real numbers) is the main example of field.

Fields I

DefinitionA field (F ,+, ·) is a 3-tuple consisting of a set F and two binary operations+, · : F × F → F such that the following nine properties hold

1 Associativity of +:

∀α, β, γ ∈ F , (α+ β) + γ = α+ (β + γ)

2 Commutativity of +:∀α, β ∈ F , α+ β = β + α

3 Existence of additive identity:

∃!0 ∈ F such that ∀α ∈ F , α+ 0 = 0+ α = α

4 Existence of additive inverse:

∀α ∈ F ∃!(−α) ∈ F such that α+ (−α) = (−α) + α = 0

Define α− β = α+ (−β)

Fields IIDefinition

5 Associativity of · :

∀α, β, γ ∈ F , (α · β) · γ = α · (β · γ)

6 Commutativity of · :∀α, β ∈ F , α · β = β · α

7 Existence of multiplicative identity:

∃!1 ∈ F s.t. 1 6= 0 and ∀α ∈ F , α · 1 = 1 · α = α

8 Existence of multiplicative inverse:

∀α ∈ F s.t. α 6= 0 ∃!α−1 ∈ F s.t. α · α−1 = α−1 · α = 1

Define αβ = αβ−1.

9 Distributivity of multiplication over addition:

∀α, β, γ ∈ F , α · (β + γ) = α · β + α · γ

Examples of Fields

Examples: FieldsUsing the standard definitions of addition and multiplication:

R is a field (verify this).Q is a field (verify this).

N is not a field: no additive identity.

Z is not a field; no multiplicative inverse for 2.

Ordered Fields

DefinitionAn ordered field is a field F which is also an ordered set such that

1 x + y < x + z if x , y , z ∈ F and y < z .2 xy > 0 if x , y ∈ F , x > 0, and y > 0.

The set of real numbers R is a field which has an order (less than or eqaul to)and another property.

The order is defined by the binary relation less than or equal to.

Axioms for RThe real numbers

R is a field (with the “usual”+ and ·, where additive identity defines 0 andmultiplicative identity defines 1) that also satisfies the following axioms.

1 Order Axiom: There is a complete ordering ≤. That is ≤ is reflexive,transitive, antisymmetric (α ≤ β, β ≤ α⇒ α = β) binary relation such that

∀α, β ∈ R either α ≤ β or β ≤ αNote, the order is compatible with + and ·, i.e.

∀α, β, γ ∈ R{

α ≤ β ⇒ α+ γ ≤ β + γα ≤ β, 0 ≤ γ ⇒ αγ ≤ βγ

(notation: α ≥ β means β ≤ α, and α < β means α ≤ β and α 6= β).2 Completeness Axiom: Suppose L,H ⊆ R, L 6= ∅ 6= H satisfy

` ≤ h ∀` ∈ L, and ∀h ∈ HThen ∃α ∈ R such that ` ≤ α ≤ h ∀` ∈ L, ∀h ∈ H

Real Numbers and Rational Numbers

Rational numbers fail completeness.

In other words, the set of rationals has “holes”.

Let A be all positive rationals p such that p2 < 2 and B consist of all positiverationals p such that p2 > 2. A contains no largest number and B contains nosmallest (this was a question in Problem Set 1).Yet, there is no rational q such that q2 = 2 (proved yesterday in class).

This is true even though given any two rationals m < n there aways existanother rational q such that m < q < n (q = m+n

2 ).

The set of real number does not have the same holes.

For every real x > 0 and every integer n > 0 there is a unique positive real ysuch that y n = x (this defines n

√x or x

1n ).

The reals fill the holes in the rationals (the rational are dense in the reals).

Upper and Lower Bounds

Since R is an ordered field, given one of its subsets we can talk about the realnumbers that are larger or lower than any element of that subset.

DefinitionGiven a set X ⊆ R,

u is an upper bound for X if

u ≥ x ∀x ∈ X

` is a lower bound for X if

` ≤ x ∀x ∈ X ;

X is bounded above if there exists an upper bound for X .

X is bounded below if there exists a lower bound for X .

Sups and Infs

One can define the smallest among the upper bounds and the largest amongthe lower bounds.

DefinitionSuppose X ⊆ R is bounded above. The supremum of X , written supX , is thesmallest upper bound for X ; that is, supX satisfies

1 supX ≥ x ∀x ∈ X supX is an upper bound2 ∀y < supX , ∃x ∈ X such that x > y there is no smaller upper bound

DefinitionSuppose X ⊆ R is bounded below. The infimum of X , written inf X , is thegreatest lower bound for X ; that is, inf X satisfies

1 inf X ≤ x ∀x ∈ X inf X is a lower bound2 ∀y > inf X , ∃x ∈ X such that x < y there is no greater lower bound

Sups and Infs

NOTATIONIf X is not bounded above, write supX =∞.If X is not bounded below, write inf X = −∞.Convention: sup ∅ = −∞, inf ∅ = +∞.

RemarksupX and inf X need not be elements of X .

Examplessup(0, 1) = 1 /∈ (0, 1)inf(0, 1) = 0 /∈ (0, 1).How about (0, 1]?

The Supremum Property

The Supremum PropertyEvery nonempty set of real numbers that is bounded above has a supremumwhich is a real number.

Every nonempty set of real numbers that is bounded below has an infimumwhich is a real number.

TheoremThe Supremum Property and the Completeness Axiom are equivalent.

This is an if and only if statement.

Proof in the next two slides.

Proof of the Supremum Property I

Proof.

Assume the Completeness Axiom and show that supX and inf X exist and are areal numbers.

Let X ⊆ R be a nonempty set that is bounded above. Let U be the set of allupper bounds for X .

Since X is bounded above, U 6= ∅. If x ∈ X and u ∈ U, x ≤ u since u is anupper bound for X .

So x ≤ u ∀x ∈ X , u ∈ U

By the Completeness Axiom,

∃α ∈ R such that x ≤ α ≤ u ∀x ∈ X , u ∈ U

α is an upper bound for X , and it is less than or equal than every other upperbound for X , so it is the least upper bound for X , so supX = α ∈ R.The case in which X is bounded below is similar. (show it)

Thus, the Supremum Property holds.

Proof of the Supremum Property IIProof.

Assume the Supremum Property and show that the completeness axiom holds.

Suppose L,H ⊆ R, L 6= ∅ 6= H, and

` ≤ h ∀` ∈ L, h ∈ H

Since L 6= ∅ and L is bounded above (by any element of H), sup L exists andit is a real number, so let α = sup L.

By the definition of supremum, α is an upper bound for L, so

` ≤ α ∀` ∈ L

Suppose h ∈ H. Then h is an upper bound for L, so by the definition ofsupremum, α ≤ h.Therefore, we have shown that

` ≤ α ≤ h ∀` ∈ L, ∀h ∈ H

so the Completeness Axiom holds.

The Archimedean Property

The Supremum Property: Every nonempty set of real numbers that is boundedabove has a supremum, which is a real number. Every nonempty set of realnumbers that is bounded below has an infimum, which is a real number.

The supremum property is useful to prove other properties of real numbers

The Archimedean Property

∀x , y ∈ R, y > 0, ∃n ∈ N such that x < ny =(y + · · ·+ y)︸ ︷︷ ︸n times

Proof.Problem Set 2.This is a proof by contradiction, using the Supremum Property.

Maximum and Minimum

DefinitionGiven a set X ⊆ R, we say a is a maximum for X (denoted maxX ) if

a ∈ X and a ≥ x ∀x ∈ X

and we say b is a minimum for X if

b ∈ X and b ≤ x ∀x ∈ X

Maximum and minimum do not always exist even if the set is bounded, butthe sup and the inf do always exist if the set is bounded.

If sup and inf are also elements of the set, then they coincide with max andmin.

Simple Result

TheoremGiven a set X ⊆ R, if maxX exists it is equal to supX.

Proof.Let a ≡ maxX . We need to show that a equals supX . To do this we must show:that (1) a is an upper bound, and (2) for every other upper bound a′, we havea′ ≥ a.

1 is easy: ∀y < supX , ∃x ∈ X such that x > y

a is an upper bound on X since a ≥ x ∀ x ∈ X by definition of maxX .

2 is also easy: let a′ 6= a be an upper bound (a′ ≥ x ∀ x ∈ X ) and supposea′ < a.

Since a ∈ X (because a = maxX ), this yields an immediate contradiction (a′

cannot be an upper bound).

Prove that if minX exists it is equal to inf X as exercise.

Intermediate Value TheoremTheorem (Intermediate Value Theorem)Let a, b ⊂ R. Suppose f : [a, b]→ R is continuous, and f (a) < d < f (b). Thenthere exists c ∈ (a, b) such that f (c) = d.

Proof: construct B = {x ∈ [a, b] : f (x) < d} and then show c = supB.

Proof of the Intermediate Value Theorem I

f : [a, b]→ R continuous with f (a) < d < f (b) ⇒ ∃c ∈ (a, b) s.t f (c) = d

This proof uses the Supremum Property.

Proof.Let

B = {x ∈ [a, b] : f (x) < d}

a ∈ B, so B 6= ∅; and B ⊆ [a, b], so B is bounded aboveBy the Supremum Property, supB exists and is a real number, so letc = supB.

Since a ∈ B, c ≥ a. B ⊆ [a, b], so c ≤ b. Therefore, c ∈ [a, b].Assume f (c) 6< d and f (c) 6> d .

Since f (c) 6< d and f (c) 6> d , we must have f (c) = d (by the order axiom).Since f (a) < d and f (b) > d , a 6= c 6= b, so c ∈ (a, b).

This finishes the proof... but we still have to show that f (c) 6< d andf (c) 6> d .

Proof of the Intermediate Value Theorem II

Proof.Need to show that f (c) 6< d . Suppose not: f (c) < d .

Since f (b) > d , we have c 6= b, and therefore c < b.Let ε = d−f (c)

2 > 0. Since f is continuous at c , there exists δ > 0 such that

|x − c | < δ ⇒ |f (x)− f (c)| < εf (x) < f (c) + ε

= f (c) + d−f (c)2

= f (c)+d2

< d+d2

= d

hence f (x) < d , and therefore (c , c + δ) ⊆ B (rememberB = {x ∈ [a, b] : f (x) < d}),this implies c 6= supB, a contradiction; therefore we have f (c) 6< d .

Proof of the Intermediate Value Theorem IIIProof.Need to show that f (c) 6> d . Suppose not: f (c) > d .

Since f (a) < d , a 6= c , so c > a.Let ε = f (c)−d

2 > 0. Since f is continuous at c , there exists δ > 0 such that

|x − c | < δ ⇒ |f (x)− f (c)| < εf (x) > f (c)− ε

= f (c)− f (c)−d2

= f (c)+d2

> d+d2

= d

so (c − δ, c + δ) ∩ B = ∅ (remember B = {x ∈ [a, b] : f (x) < d}).Thus, either there exists x ∈ B with x ≥ c + δ (in which case c is not anupper bound for B) or c − δ is an upper bound for B (in which case c is notthe least upper bound for B);

in either case, c 6= supB, a contradiction; thus f (c) 6> d .

Implications of the Intermediate Value TheoremThe intermediate value theorem can be used in all sorts of ways

CorollaryThere exists x ∈ R such that x2 = 2.

Proof.Let f (x) = x2, for x ∈ [0, 2].

f is continuous (we proved this yesterday).

f (0) = 0 < 2 and f (2) = 4 > 2,

so by the Intermediate Value Theorem, there exists c ∈ (0, 2) such thatf (c) = 2, i.e. such that c2 = 2.

In Problem Set, you have to show (wihout cheating) that for every realnumber x > 0 and every integer n > 0 there is a unique positive real numbery such that yn = x .

This defines n√x or x

1n .

Vector SpaceNow that we have defined a good number concept, we define useful propertiesof the set that collects the objects of interest.

This set should have (at least) two minimal properties:

one should be able to add two of its elements, andone should be able to scale up and down any of its elements.

These properties are useful to give meaning to the idea that elements of a setare large or small, and close or far.

A vector (or linear) space is a set with linear structure and where some basicoperations are possible.

The basic operations are addition and multiplication by a scalar (an element ofa field), and one wants them to have the usual properies.Linear structure means that the sum of any two objects is also an element ofthe set (closure under addition), and that a scalar multiple of any object is alsoin the set (closure under multiplication by a scalar ).The set should contain a “zero” element, and multiplying any element by thescalar one should not change it.

Elements of a vector space are “vector-like”.

Rn (the set of n-dimensional real numbers) is the main example of vector space.

Vector Spaces I

DefinitionA vector space V is a 4-tuple (V ,F ,+, ·) where V is a set of elements, calledvectors, F is a field, a binary operation + : V × V → V called vector addition, andan operation · : F × V → V called scalar multiplication, satisfying

1 Closure under addition

∀x, y ∈ V (x+ y) ∈ V

2 Associativity of +:

∀x, y, z ∈ V (x+ y) + z = x+ (y + z)

3 Commutativity of +:

∀x, y ∈ V x+ y = y + x

4 Existence of vector additive identity:

∃!0 ∈ V such that ∀x ∈ V , x+ 0 = 0+ x = x

Vector Spaces IIDefinition

5 Existence of vector additive inverse:

∀x ∈ V ∃!y ∈ V such that x+ y = y + x = 0

We define (−x) to be this unique y (and let x− y be x+ (−y)).6 Distributivity of scalar multiplication over vector addition:

∀α ∈ F , ∀x, y ∈ V α · (x+ y) = α · x+ α · y

7 Distributivity of scalar multiplication over scalar addition:

∀α, β ∈ F , ∀x ∈ V (α+ β) · x = α · x+ β · x

8 Associativity of · :

∀α, β ∈ F , ∀x ∈ V (α · β) · x = α · (β · x)

9 Multiplicative identity:∀x ∈ V 1 · x = x

SubspacesIn this class, vector spaces are always over real numbers.

DefinitionA vector space V over R is (V ,R,+, ·) where V is a set of vectors,+ : V × V → V is vector addition, · : R× V → V is scalar multiplication, s.t.

1 Closure under addition: the sum of two vectors is a vector.2 Associativity of +.3 Commutativity of +.4 There exists a unique vector we call zero.5 There exists a unique ‘negative’of any vector.6 Distributivity of scalar multiplication over vector addition (the scalar in R).7 Distributivity of scalar multiplication over scalar addition (the scalar in R).8 Associativity of · (the scalar in R).9 The real number 1 multiplied by a vector yields that same vector (uses R).

DefinitionA subset of a vector space that is itself a vector space is called a subspace.

Euclidean SpacesDefinitionFor each positive integer n the corresponding Euclidean space, denoted Rn , is theset of all ordered n-tuples

x = (x1, x2, ..., xn)

where each of the x1, ..., xn are real numbers (sometimes called coordinates).

The elements of Rn are vectors (or points).

NotationIf x and y are vectors and if α is a real number:

x+ y = (x1 + y1, x2 + y2, ..., xn + yn)

αx = (αx1, αx2, ..., αxn)

The zero element of Rn is the vector

0 = (0, ..., 0)

Examples of Vector Spaces

Euclidean Spaces are Vector SpacesR and Rn are vector spaces (verify this).

The Set of All Real Valued Functions is a Vector SpaceLet X be a non empty set; the set of all functions f : X → R is a vector space.

vector addition works since we defined sums of functions as:(f + g)(x) = f (x) + g(x)

scalar multiplication also works since we defined as: (αf )(x) = α(f (x))

vector additive identity: 0 is the function which is identically zero (f (x) = 0for any x).

vector additive inverse: (−f )(x) = −(f (x))

NOTEIf we take X = {1, 2, ..., n}, this is the set of all vectors in Rn .If we take X = {1, 2, ...}, this is the set of all infinite real sequences.

Tomorrow

Introduce the idea of distance between two elements of a set; introduce theidea of length of a vector, and look at special vector spaces where length is awell defined concept; focus on sets where distance coincides with the length ofthe difference; sequences and limits are defined using thse concepts.

1 Metric and Metric Spaces2 Norm and Normed Spaces3 Sequences and Subsequences4 Convergence5 Monotone and Bounded Sequences