An Introduction to Proof via Inquiry-Based Learning

We begin with the Field Axioms.

In the language of abstract algebra, Axioms F1–F4 and F5–F8 make each of \(\mathbb{R}\) and \(\mathbb{R}\setminus\{0\}\) an abelian group under addition and multiplication, respectively. Axiom F9 provides a way for the operations of addition and multiplication to interact. Collectively, Axioms F1–F9 make the real numbers a field. Axioms F3 and F7 state the existence of additive and multiplicative identities, but these axioms do not assume that the elements are the unique elements with the specified properties. However, we can prove that this is the case. That is, \(0\) and \(1\) of \(\mathbb{R}\) are the unique additive and multiplicative identities in \(\mathbb{R}\text{.}\) To prove the following theorem, suppose \(0\) and \(0'\) are both additive identities in \(\mathbb{R}\) and then show that \(0=0'\text{.}\) This shows that there can only be one additive identity. It is important to point out that we are not proving that the number \(0\) introduced in Axiom F3 is unique, but rather there is a unique number with the property specified in Axiom F3.

To prove the next theorem, mimic the approach you used to prove Theorem 5.2.

Similar to Axioms F3 and F7, Axioms F4 and F8 state the existence of additive and multiplicative inverses, but these axioms do not assume that these elements are the unique elements with the specified properties. However, we can prove that for every \(a\in\mathbb{R}\text{,}\) the elements \(-a\) and \(a^{-1}\) (as long as \(a\neq 0\)) are the unique additive and multiplicative inverses, respectively.

In light of the last two theorems, we now know that sticking a minus sign in front of \(a\in\mathbb{R}\) or raising \(a\in\mathbb{R}\setminus\{0\}\) to \(-1\) each correspond to an operation that yields a unique element with the corresponding inverse property. Note that since \(0+0=0$\)and additive inverses are unique, it must be the case that \(-0=0\text{.}\)

Since we are taking a formal axiomatic approach to the real numbers, we should make it clear how the natural numbers are embedded in \(\mathbb{R}\text{.}\)

Notice the similarity between the definition of the natural numbers presented above and the Axiom of Induction given in Section 4.1. Of course, we use the standard numeral system to represent the natural numbers, so that \(\mathbb{N}= \{1,2,3,4,5,6,7,8,9,10\ldots\}\text{.}\)

Given the natural numbers, Axiom F3/Theorem 5.2 and Axiom F4/Theorem 5.4 together with the operation of addition allow us to define the integers, denoted by \(\mathbb{Z}\text{,}\) in the obvious way. That is, the integers consist of the natural numbers together with the additive identity and all of the additive inverses of the natural numbers.

We now introduce some common notation that you are likely familiar with. Take a moment to think about why the following is a definition as opposed to an axiom or theorem.

Using the Field Axioms, we can prove each of the statements in the following theorem.

Carefully prove the next theorem by explicitly citing where you are utilizing the Field Axioms and Theorem 5.8.

We now introduce the Order Axioms of the real numbers.

Given Axioms O1–O4, we say that the real numbers are a linearly ordered field. We call numbers greater than zero positive and those greater than or equal to zero nonnegative. There are similar definitions for negative and nonpositive.

Notice that the Order Axioms are phrased in terms of “\(\lt\)”. We would also like to be able to utilize “\(>\)”, “\(\leq\)”, and “\(\geq\)”.

Notice that we took the existence of the inequalities “\(\lt\)”, “\(>\)”, “\(\leq\)”, and “\(\geq\)” on the real numbers for granted when we defined intervals of real numbers in Definition 3.4.

Using the Order Axioms, we can prove many familiar facts.

The next result extends Axiom O3.

Consider using three cases when approaching the proof of the following theorem.

It might come as a surprise that the following result requires proof.

The previous theorem together with Theorem 5.14 implies that \(-1\lt 0\) as you expect. It also follows from Axiom O3 that for all \(n\in\mathbb{Z}\text{,}\) we have \(n\lt n+1\text{.}\) We assume that there are no integers between \(n\) and \(n+1\text{.}\)

The last few results allow us to take for granted our usual understanding of which real numbers are positive and which are negative. The next theorem yields a result that extends Theorem 5.21.

There is a special function that we can now introduce.

We can interpret \(|a|\) as the distance between \(a\) and 0 as depicted in Figure 5.1.

Given two points \(a\) and \(b\text{,}\) \(|a-b|\text{,}\) and hence \(|b-a|\) by the previous theorem, is the distance between \(a\) and \(b\) as shown in Figure 5.2.

In the next theorem, writing \(\pm a\leq b\) is an abbreviation for \(a\leq b\) and \(-a\leq b\text{.}\)

The letter \(r\) was used in the previous theorem because it is the first letter of the word “radius”. If \(r\) is positive, we can think of the interval \((-r,r)\) as the interior of a one-dimensional circle with radius \(r\) centered at 0. Figure 5.3 provides a visual interpretation of Theorem 5.30.

Since \(|a-b|\) represents the distance between \(a\) and \(b\text{,}\) we can interpret \(|a-b|\leq r\) as saying that the distance between \(a\) and \(b\) is less than or equal to \(r\text{.}\) In other words, \(a\) is within \(r\) units of \(b\text{.}\) See Figure 5.4.

Consider using Theorem 5.29 and Theorem 5.30 when attacking the next result, which is known as the Triangle Inequality. This result can be extremely useful in some contexts.

Figure 5.5 depicts two of the cases for the Triangle Inequality.

Where did the Triangle Inequality get its name? Why “Triangle”? For any triangle (including degenerate triangles), the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. That is, if \(x\text{,}\) \(y\text{,}\) and \(z\) are the lengths of the sides of the triangle, then \(z\leq x+y\text{,}\) where we have equality only in the degenerate case of a triangle with no area. In linear algebra, the Triangle Inequality is a theorem about lengths of vectors. If \(\mathbf{a}\) and \(\mathbf{b}\) are vectors in \(\mathbb{R}^n\text{,}\) then the Triangle Inequality states that \(\lVert \mathbf{a}+\mathbf{b}\rVert \leq \lVert\mathbf{a}\rVert +\lVert\mathbf{b}\rVert\text{.}\) Note that \(\lVert \mathbf{a}\rVert\) denotes the length of vector \(\mathbf{a}\text{.}\) See Figure 5.6. The version of the Triangle Inequality that we presented in Theorem 5.32 is precisely the one-dimensional version of the Triangle Inequality in terms of vectors.

The next theorem is sometimes called the Reverse Triangle Inequality.

Before we introduce the Completeness Axiom, we need some additional terminology.

Notice that a set \(A\subseteq\mathbb{R}\) is bounded if and only if it is a subset of some bounded closed interval.

Our next result tells us that a supremum of a set and an infimum of a set are unique when they exist.

In light of the previous theorem, if the supremum of \(A\) exists, it is denoted by \(\tcboxmath{\sup(A)}\text{.}\) Similarly, if the infimum of \(A\) exists, it is denoted by \(\tcboxmath{\inf(A)}\text{.}\)

It is important to recognize that the supremum or infimum of a set may or may not be contained in the set. In particular, we have the following theorem concerning suprema and maximums. The analogous result holds for infima and minimums.

Intuitively, a point is the supremum of a set \(A\) if and only if no point smaller than the supremum can be an upper bound of \(A\text{.}\) The next result makes this more precise.

The following axiom states that every nonempty subset of the real numbers that has an upper bound has a least upper bound.

Given the Completeness Axiom, we say that the real numbers satisfy the least upper bound property. It is worth mentioning that we do not need the Completeness Axiom to conclude that every nonempty subset of the integers that is bounded above has a supremum, as this follows from Theorem 4.40 (a generalized version of the Well-Ordering Principle).

Certainly, the real numbers also satisfy the analogous result involving infimum.

Our next result, called the Archimedean Property, tells us that for every real number, we can always find a natural number that is larger. To prove this theorem, consider a proof by contradiction and then utilize the Completeness Axiom and Theorem 5.43.

More generally, we can “squeeze” every real number between a pair of integers. The next result is sometimes referred to at the Generalized Archimedean Property.

The next theorem strengthens the Generalized Archimedean Property and says that every real number is either an integer or lies between a pair of consecutive integers. To prove this theorem, let \(x\in\mathbb{R}\) and define \(L=\{k\in\mathbb{Z}\mid k\leq x\}\text{.}\) Use the Generalized Archimedean Property to conclude that \(L\) is nonempty and then utilize Theorem 4.40.

To prove the next theorem, let \(a\lt b\text{,}\) utilize Theorem 5.49 on \(b-a\) to obtain \(N\in\mathbb{N}\) such that \(\frac{1}{N}\lt b-a\text{,}\) and then apply Theorem 5.50 to \(Na\) to conclude that there exists \(n\in\mathbb{N}\) such that \(n\leq Na\lt n+1\text{.}\) Lastly, argue that \(\frac{n+1}{N}\) is the rational number you seek.

Recall that the real numbers consist of rational and irrational numbers. Two examples of an irrational number that you are likely familiar with are \(\pi\) and \(\sqrt{2}\text{.}\) In Section 6.2, we will prove that \(\sqrt{2}\) is irrational, but for now we will take this fact for granted. It turns out that \(\sqrt{2}\approx 1.41421356237\in (1,2)\text{.}\) This provides an example of an irrational number occurring between a pair of distinct rational numbers. The following theorem is a good challenge to generalize this.

Repeated applications of the previous two theorems implies that every open interval contains infinitely many rational numbers and infinitely many irrational numbers. In light of these two theorems, we say that both the rationals and irrationals are dense in the real numbers.