One of the central questions explored in this class revolves around the concept of a real number. A technical satisfactory definition of a real number is the following: A real number is the limit of a sequence of rational numbers forming a Cauchy sequence. For the novice real analyst, this definition is perhaps a bit too mouthful. One may ask: What exactly is a rational number? And what is meant by a sequence of rational numbers? What about the concept of limit? What exactly is this notion aiming to capture? Also what is the precise definition of a Cauchy sequence? Although these terms may have been encountered in previous classes, at least superficially, a complete understanding may not have been fully achieved.

Ultimately, one of the main objectives of this course is to reach a point where we are able to define in precise terms what a real number is. To accomplish this, we will provide a clear definition of a sequence, a rational number, a limit, and the characteristics of Cauchy sequences. Our intention is to equip the reader with the necessary language and tools needed to grasp this rather technical definition of a real number.

Let’s begin by defining what a rational number is. We will not attempt to be exhaustive in this endeavor. Rather, our goal is to outline a construction that aligns with the understanding of rational numbers we’ve already developed in the earlier years of our education. This approach will offer a theoretical foundation with an added level of rigor. To do so, we’ll start by defining the set of natural numbers. Informally, the set of natural numbers encompasses the numbers we use for counting, such as those we could count on our fingers. Imagine having a hand with infinitely many fingers. To count these finger, we would start with, maybe no finger, one finger, then move to two, three, four, and so on. It is evident that one finger is distinct from two fingers, and two fingers are distinct from three, and so forth.

Let us now formalize this concept.

Construction of Natural Numbers

Intuition: The inductive step initiates a domino effect. The property is shown to hold true for 0 (base case) and then for 1, 2, 3, and so on, covering all natural numbers in an infinite chain. Induction can be understood as a process akin to climbing a ladder. The base case is about verifying the safety of the first rung of the ladder. The inductive step demonstrates that if any rung of the ladder is safe, the next rung will also be safe. With these two components, we can ascend the ladder safely, step by step, indefinitely.

1. a + 0 = a 
2. a+S(b)=S(a+b)

Calculating 1+1: Note that By 1, we mean 1:=S(0). As such, 1+1=S(0)+S(0)=S(S(0)+0)=S(S(0)):=2.

For instance, to compute 2+3, we may proceed as follows: since 3=S(S(S(0))), the following is immediate: 
2+3 := 2+S(S(S(0)))
     = S(2+S(S(0)))
     = S(S(2+S(0)))
     = S(S(S(2+0)))
     = S(S(S(2))) (Apply the base case)
    := S(S(3))
    := S(4)
    := 5.

Homework

We define addition recursively as follows:
a+0  : =a
a+S(b) = S(a+b)

[Lemma] For any a∈ℕ,0+a=a.

Proof: We proceed by induction. For the base case, it is clear that 0+0=0 since a+0:=a for every natural number. Now suppose that 0+a=a. We wish to establish that 0+S(a)=S(a). Since S(a)=S(0+a)=0+S(a) this completes the proof.

[Lemma] For any natural number a,b, S(a)+b=a+S(b).

Proof: We will proceed by induction on b. For the base case, S(a)+0=S(a)=S(a+0)=a+S(0). Next, suppose that S(a)+b=a+S(b). Then S(a)+S(b) = S(S(a)+b) = S(a+S(b)) = a+S(S(b)).

In light of the results above, we are now ready to establish that addition is commutative.

[Theorem] For any natural number a,b, a+b=b+a.

Proof: We will proceed by induction on b. For the base case, a+0=0+a=a. Next, let us assume that a+b=b+a. We aim to show that S(a)+b=b+S(a). It is clear that by definition, b+S(a) = S(b+a) = S(a+b) = a+S(b). By the previous lemma, it follows that b+S(a)=a+S(b)=S(a)+b and this completes the proof.

[Theorem] Addition of natural numbers is an associative operation. 

To demonstrate the associativity of addition, we can establish that for any natural numbers a, b, and c, the equation (a+b)+c=a+(b+c) holds true.

We will proceed with a proof by induction on b. For the base case, we have (a+0)+c=a+c=a+(0+c). Next, assuming that (a+b)+c=a+(b+c), we can derive the following: (a+S(b))+c = S(a+b)+c= c+S(a+b)= S(c+a+b)= S(a+c+b)= a+S(c+b)= a+(c+S(b))= a+(S(b)+c).

Multiplication

We may define multiplication of natural numbers recursively as follows: The The product of any natural number n and zero is zero, and recursively, the product of a natural number n and the successor of another natural number m denoted by S(m) is defined as the sum of n and the product of n and m. That is,

n×0  = 0
n×S(m) = n+(n×m)

The following is left as an exercise to the reader: show by induction that multiplication is a commutative as well as an associative operation.

Prove that for any natural number n ≥ 1, the sum of 3 raised to k from k equals 1 to n is equal to (3/2)3ⁿ − (3/2).

Construction of the Integers

Perhaps the best way to model negative numbers is to use a system where one can remove more units than the total at one’s disposal. This establishes a simple model in some abstract economy in which, when repaying debts, an excess amount is subtracted from what was already given. Inherently, such a model can be represented by ordered pairs, where the first element consists of the number of units in possession of the individual, and the second element represents the quantity to be removed. For example, an ordered pair consisting of (2, 2) would essentially mean that an individual has no units left in possession. Meanwhile, an ordered pair of (5, 3) would imply that the individual has essentially two units remaining. Conversely, an ordered pair of (3, 5) would indicate that the individual owes two units.

In this section, we will explore the construction of the set of integers, expanding our understanding beyond natural numbers. The set of integers not only includes the natural numbers (but also zero, regardless of the convention for the natural numbers.) Typically, this set is denoted by ℤ, derived from the German word “Zahlen,” which means numbers. Recall that zero may or may not be considered part of the natural numbers, depending on how the set of natural numbers was initially defined, and negative numbers are introduced as counterparts to natural numbers.

Ordered Pairs and Equivalence Classes

To represent integers, we use ordered pairs of natural numbers, where each pair (a, b) represents the integer a – b. For example, the ordered pair (3, 2) represents 1 in our usual understanding of integers, and the ordered pair (2, 3) represents its opposite. However, we avoid using direct subtraction since it is not yet defined. We define an equivalence relation on these pairs to ensure that each integer has a unique representation. We say that the ordered pairs (a, b) and (c, d) are equivalent if a + d = b + c in the set of natural numbers. This relation is reflexive, symmetric, and transitive, fulfilling the requirements of an equivalence relation.

• Reflexivity: A relation R on a set A is said to be reflexive if every element is related to itself. In other words, for any element a in A, aRa holds.
• Symmetry: A relation R on a set A is symmetric if whenever an element a is related to an element b, b is also related to a. Formally, if aRb, then bRa. This property implies that the relation goes both ways.
• Transitivity: A relation R on a set A is transitive if whenever an element a is related to some element b and b is related to c, then a is related to c. This property links elements in a chain-like manner.

Within this framework, we define 0 as the equivalence class of all pairs (n, n), where n is a natural number: 0 = [(1, 1)] = {(1, 1), (2, 2), (3, 3), (4, 4), …}. Next, negative numbers are the equivalence classes of pairs where the second component is greater than the first, such as (2, 3), and [(2, 3)] = {(3, 4), (4, 5), …} = {(n, n+1): n is a natural number} represents the class of integers equivalent to (2, 3).

Addition and Multiplication

(a, b) + (c, d) = (a + c, b + d)

To add two integers represented by the ordered pairs (a, b) and (c, d), we simply add their components: (a, b) + (c, d) = (a + c, b + d). Can you see why this definition of addition is consistent with our expectations?

(1,1) is the neutral element for addition of integers viewed as classes of equivalent ordered natural numbers. 

Homework

Multiplication of integers is defined as follows: given ordered pairs (a, b) and (c, d), their product is given by the pair (ac + bd, ad + bc).

Homework

Both addition and multiplication operations are closed since the sum and product of any two pairs (a, b) and (c, d) are also pairs that represent integers. Hence, the set of integers is closed under addition and multiplication.

Homework

Although we shall not prove this fact, it is worth noting that addition and multiplication of integers are defined earlier are commutative as well as associative operations. This exercise is left to the reader.

Visualizing Integers

Construction of the Rational Numbers ℚ

Defining Rational Numbers

The highschool definition of a rational number is something along the following lines: a rational number is a number that can be expressed as a fraction of the form a/b, where a and b are integers and b is not equal to zero. This representation allows us to express a wide range of numbers, including natural and integral numbers. Perhaps, a rigorous way to view rational numbers is as elements of equivalence classes as well.

We define rational numbers as equivalence classes of ordered pairs of integers (a, b), and we say that two pairs of integers (a, b) and (c, d) are equivalent if ad = bc. 

For instance, let us consider the pairs (1, 2) and (2, 4). These two pairs are equivalent to each other since 1*4 = 2*2 = 4.

Conventionally, we shall denote a pair of the form (a, b) by the notation or symbol a/b. We define addition and multiplication of rational numbers via their representatives:

• Addition: a/b + c/d = (ad + bc) / (b*d)
• Multiplication: a/b * c/d = (ac) / (bd)

Consider the set of ordered pairs (a, b) where a and b are integers and b is nonzero. Each pair can be thought of as a fraction of the form a/b. An equivalence relation ~ is defined on the set of these ordered pairs, where (a, b) ~ (c, d) if and only if ad = bc. This relation is reflexive, symmetric, and transitive.

• Reflexive: (a, b) ~ (a, b) because ab = ba (commutativity of multiplication).
• Symmetric: If (a, b) ~ (c, d), then (c, d) ~ (a, b). Indeed, by assuming that (a, b) ~ (c, d), we have ad = bc, and this implies that cb = da, showing that (c, d) ~ (a, b).
• Transitive: If (a, b) ~ (c, d) and (c, d) ~ (e, f), then (a, b) ~ (e, f). Indeed, by assumption, we have ad = bc and cf = de, and we seek to establish that af = be.

The image below depicts rational numbers as groups of ordered pairs. Each point in the picture symbolizes a rational number. Rational numbers that are equivalent, or have the same value, are marked with identical colors, illustrating their equivalence.

Arithmetic on Rational Numbers

Arithmetic on rational numbers involves the operations of addition, subtraction, multiplication, and division, similar to those on integers but with an added layer of complexity due to the presence of denominators.

• Closure: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except by zero).
• Commutativity and Associativity: Rational number arithmetic is commutative and associative for both addition and multiplication.
• Existence of Identity Elements: The additive identity is 0 (represented by 0/1), and the multiplicative identity is 1 (represented by 1/1).

Ordering Structure of Rational Numbers

The ordering structure of the rational numbers gives us the ability to compare two distinct rational numbers, as it is often useful to be able to determine when one rational number is ‘larger’ than another one. 

As a starter, we will establish an ordering structure on the integers

For any integers a and b, we say that a<b if there exists a positive integer (a natural number that is not zero) p such that a+p=b. 

The following holds true

1.[Transitivity] If a<b and b<c then a<c.
2.[Antisymmetry] If a<b and b<a then a=b
3.[Totality] For any integers a and b one and only one of the following remains true: a<b,a=b or a>b.

For the first claim, assume that a<b and b<c. Then there exist natural numbers p,q such that a+p=b and b+q=c. Consequently, b+q=a+p+q=a+(p+q)=c. Since p+q∈ℕ it follows that a<c. For the second claim, assume that a<b and b<a. Then there exist natural numbers p,q such that a+p=b,b+q=a. Consequently, a+p=a and p=0 and a=b. The proof of the third part is left as an exercise.

The following properties also hold true:

∙ (Addition preserves order) If a>b then a+c>b+c
∙ (Positive multiplication preserves order) If a>b and c>0 then ac>bc
∙ (Negation reverses order) If a>b then -a<-b

We will now extend this ordering structure to the rational numbers.

Given two rational numbers a/b and c/d we say that a/b<c/d if and only if ad<bc assuming that b,d are positive integers. 

The 3 properties established above are naturally extended to the rational numbers. That is:

1. [Transitivity] Given a,b,c∈ℚ, if a<b and b<c then a<c.
2. [Antisymmetry] Given a,b∈ℚ if a<b and b<a then a=b.
3. [Totality] For any rational numbers a and b one and only one of the following holds true: a<b, a=b or a>b

The following properties also hold true:

∙ (Addition preserves order) If a>b then a+c>b+c
∙ (Positive multiplication preserves order) If a>b and c>0 then ac>bc
∙ (Negation reverses order) If a>b then -a<-b

Absolute value

Let x be a rational number. The absolute value of x, denoted by |x|, is defined as follows: |x| is equal to x if x is nonnegative (x≥0) and is equal to its opposite otherwise (if x≤0.)

Next, we define the distance function using the absolute value. Let x and y be real numbers. The distance between x and y is defined as the absolute value of their difference, |x – y|.

The non-degeneracy property of the absolute value states that the absolute value of a number is equal to zero if and only if the number is zero. It can be proven by considering the cases where the number is positive, negative, or zero.

Triangle Inequality

The Triangle Inequality is a fundamental property of the absolute value. For any rational numbers x and y, it states that |x + y| is less than or equal to |x| + |y|.