Chapter 4 The real numbers
4.1 Cauchy sequences
Definition 4.1 (Sequences) Let \(m\) be an integer. A sequence \((a_n)_{n=m}^{\infty}\) of rational numbers is ant function from the set \(\{n\in \mathbb{Z}: n \geq m\) to \(\mathbb{Q}\), i.e., a mapping which assigns to each integer \(n\) greater than or equal to \(m\), a rational number \(a_n\). More informally, a sequence \((a_n)_{n=m}^{\infty}\) of rational numbers is a collection of rationals \(a_m, a_{m+1}, a_{m+2}, \dots\)
Definition 4.2 ($\varepsilon$-steadness) Let \(\varepsilon>0\). A sequence \((a_n)_{n=0}^{\infty}\) is said to be \(\varepsilon\)-steady iff each pair \(a_j, a_k\) of sequence elements is \(\varepsilon\)-close for every natural number \(j,k\). In order words, the sequence \(a_0,a_1,\dots\) is \(\varepsilon\)-steady iff \(d(a_j,a_k)\leq \varepsilon\) for all \(j,k\).
Definition 4.3 (Eventual $\varepsilon$-steadness) Let \(\varepsilon>0\). A sequence \((a_n)_{n=0}^{\infty}\) is said to be eventually \(\varepsilon\)-steady iff the sequence \(a_N, a_{N+1}, a_{N+2}, \dots\) is \(\varepsilon\)-steady for some natural number \(N\geq 0\). In other words, the sequence \(a_0,a_1,\dots\) is eventually \(\varepsilon\)-steady iff there exists an \(N\geq 0\) such that \(d(a_j,a_k)\leq \varepsilon\) for all \(j,k\geq N\).
Definition 4.4 (Cauchy sequences) A sequence \((a_n)_{n=0}^{\infty}\) of rational numbers is said to be Cauchy sequences iff for every rational \(\varepsilon>0\), thesequence \((a_n)_{n=0}^{\infty}\) is eventually \(\varepsilon\)-steady.
Definition 4.5 (Bounded sequences) Let \(M\geq 0\) be rational. A finite sequence \(a_1,a_2, \dots, a_n\) is bounded by \(M\) iff \(|a_i|\leq M\) for all \(1\leq i \leq n\). An infinite sequence \((a_n)_{n=1}^{\infty}\) is bounded by \(M\) iff iff \(|a_i|\leq M\) for all \(i\geq 1\). A sequence is said to be bounded iff it is bounded by \(M\) for some rational \(M>0\).
Lemma 4.1 Every finite sequence is bounded.
Lemma 4.2 Every Cauchy sequence \((a_n)_{n=1}^{\infty}\) is bounded.
4.2 Equivalent Cauchy sequences
Definition 4.6 ($\varepsilon$-close sequences) Let \((a_n)_{n=0}^{\infty}\) and \((b_n)_{n=0}^{\infty}\) be two sequences, and let \(\varepsilon>0\). We say the sequence \((a_n)_{n=0}^{\infty}\) is \(\varepsilon\)-close to \((b_n)_{n=0}^{\infty}\) iff \(a_n\) is \(\varepsilon\)-close to \(b_n\) for each \(n\in \mathbb{N}\). In other wors, the sequence \((a_n)_{n=0}^{\infty}\) is \(\varepsilon\)-close to \((b_n)_{n=0}^{\infty}\) iff \(|a_n-b_n|\leq \varepsilon\) for all \(n=0,1,2,\dots\).
Definition 4.7 ($Eventually \varepsilon$-close sequences) Let \((a_n)_{n=0}^{\infty}\) and \((b_n)_{n=0}^{\infty}\) be two sequences, and let \(\varepsilon>0\). We say the sequence \((a_n)_{n=0}^{\infty}\) is eventually \(\varepsilon\)-close to \((b_n)_{n=0}^{\infty}\) iff there exists an \(N>0\) such that the sequences \((a_n)_{n=N}^{\infty}\) and \((b_n)_{n=N}^{\infty}\) are \(\varepsilon\)-close.
Definition 4.8 (Equivalent sequences) Two sequences \((a_n)_{n=0}^{\infty}\) and \((b_n)_{n=0}^{\infty}\) are equivalent iff for each rational \(\varepsilon >0\), the sequences \((a_n)_{n=0}^{\infty}\) and \((b_n)_{n=0}^{\infty}\) are eventually \(\varepsilon\)-close.
4.3 The construction of the real numbers
Definition 4.9 (Real numbers) A real number is defined to be an object of the form \({\rm LIM}_{n\to \infty}a_n\), where \((a_n)_{n=1}^{\infty}\) is a Cauchy sequence of rational numbers. Two real numbers \({\rm LIM}_{n\to \infty}a_n\) and \({\rm LIM}_{n\to \infty}b_n\) are said to be equal iff \((a_n)_{n=1}^{\infty}\) and \((b_n)_{n=1}^{\infty}\) are equivalent Cauchy sequences. The set of all real numbers is denoted \(\mathbb{R}\).
From now on we will refer to \({\rm LIM}_{n\to \infty}a_n\) as the formal limit of the sequence \((a_n)_{n=1}^{\infty}\).
Proposition 4.1 (Formal limits are well-defined) Formal limits obeys reflexive axiom, symmetry axiom and transitive axiom (see Chapter 11).
Definition 4.10 (Addition of reals) Let \(x={\rm LIM}_{n\to \infty}a_n\) and \(y={\rm LIM}_{n\to \infty}b_n\) be real numbers. Then we define the sum \(x+y\) to be \(x+y:{\rm LIM}_{n\to \infty}(a_n+b_n)\).
Lemma 4.3 Let \(x={\rm LIM}_{n\to \infty}a_n\) and \(y={\rm LIM}_{n\to \infty}b_n\) be real numbers. Then \(x+y\) is also a real number.
Lemma 4.4 Let \(x={\rm LIM}_{n\to \infty}a_n\), \(y={\rm LIM}_{n\to \infty}b_n\) and \(x'={\rm LIM}_{n\to \infty}a_n'\) be real numbers. Suppose that \(x=x'\). Then we have \(x+y=x'+y\).
The above lemma verify the axiom of substitution.
Definition 4.11 (Multiplication of reals) Let \(x={\rm LIM}_{n\to \infty}a_n\) and \(y={\rm LIM}_{n\to \infty}b_n\) be real numbers. Then we define the product \(xy\) to be \(xy:{\rm LIM}_{n\to \infty}a_n b_n\).
Lemma 4.5 Let \(x={\rm LIM}_{n\to \infty}a_n\), \(y={\rm LIM}_{n\to \infty}b_n\) and \(x'={\rm LIM}_{n\to \infty}a_n'\) be real numbers. The \(xy\) is also a real number. Furthermore, if \(x=x'\), then \(xy=x'y\).
At this point we embed the rationals back into the reals, by equating every rational number \(q\) with the real number \({\rm LIM}_{n\to \infty}q\).
We can now easily define negation \(-x\) for real numbers \(x\) by the formula \[ -x:=(-1)\times x, \] since \(-1\) is a rational number and is hence real. Also, from our define it is clear that \[ -{\rm LIM}_{n\to \infty}a_n = {\rm LIM}_{n\to \infty}(-a_n) \]
Once we have addition and negation, we can define substraction as usual by \[ x-y:= x+(-y) \] note that this implies \[ {\rm LIM}_{n\to \infty}a_n-{\rm LIM}_{n\to \infty}b_n = {\rm LIM}_{n\to \infty}(a_n-b_n) \]
Proposition 4.2 All the laws of algebra from Proposition 4.1 hold not only for integers, but for the reals as well.
The last basic arithmetic operation we need to define is reciprocation.
Definition 4.12 (Sequences bounded away from zero) A sequence \((a_n)_{n=1}^{\infty}\) of rational numbers is said to bounded away from zero iff there exists a rational numbers \(c>0\) such that \(|a_n|>c\) for all \(n\geq 1\).
Lemma 4.6 Let \(x\) be a non-zero real number. Then \(x={\rm LIM}_{n\to \infty}a_n\) for some Cauchy sequence \((a_n)_{n=1}^{\infty}\) which is bounded away from zero.
Lemma 4.7 Suppose that \((a_n)_{n=1}^{\infty}\) is a Cauchy sequence which is bounded away from zero. Then the sequence \((a_n^{-1})_{n=1}^{\infty}\) is also a Cauchy sequence.
Definition 4.13 (Reciprocals of real numbers) Let \(x\) be a non-zero real number. Let \((a_n)_{n=1}^{\infty}\) be a Cauchy sequence bounded away from zero such that \(x={\rm LIM}_{n\to \infty}a_n\). Then we define the reciprocal \(x^{_1}\) by the formula \(x^{_1}:={\rm LIM}_{n\to \infty}a_n^{-1}\)
Lemma 4.8 (Reciprocal is well-defined) Let \((a_n)_{n=1}^{\infty}\) and \((b_n)_{n=1}^{\infty}\) be two Cauchy Sequences bounded away from zero such that \({\rm LIM}_{n\to \infty}a_n={\rm LIM}_{n\to \infty}b_n\). Then \({\rm LIM}_{n\to \infty}a^{-1}_n={\rm LIM}_{n\to \infty}b^{-1}_n\).
Once one has reciprocal, one can define division \(x/y\) of two real numbers \(x,y\), provided \(y\) non-zeo, by the formula \[ x/y:=x\times y^{-1} \]
Exercise 4.1 Let \(a,b\) be rational numbers. Show that \(a=b\) iff \({\rm LIM}_{n\to \infty}a= {\rm LIM}_{n\to \infty}b\). This allows us to embed the rational numbers inside the real number in a well-defined manner.
4.4 Ordering the reals
Definition 4.14 Let \((a_n)_{n=1}^{\infty}\) be a sequence of rationals. We say that this sequence is positive bounded away from zero iff we have positive rational \(c>0\) such that \(a_n\geq c\) for all \(n\geq 1\). The sequence is negative bounded away from zero iff we have positive rational \(c>0\) such that \(a_n\leq -c\) for all \(n\geq 1\).
Definition 4.15 A real number \(x\) is said to be positive iff it can be written as \(x={\rm LIM}_{n\to \infty}a_n\) for some Cauchy sequence \((a_n)_{n=1}^{\infty}\) which is positively bounded away from zero. \(x\) is said to be negative iff it can be written as \(x={\rm LIM}_{n\to \infty}a_n\) for some Cauchy sequence \((a_n)_{n=1}^{\infty}\) which is negatively bounded away from zero.
Proposition 4.3 (Basic properties of positive reals) For every real number \(x\), exactly one of the following three statements is true: (a) \(x\) is zero; (b) \(x\) is positive; (c) \(x\) is negative. A real number \(x\) is negative if and only if \(-x\) is positive. If \(x\) and \(y\) are positive, then so are \(x+y\) and \(xy\).
One can prove this with 4.6.
Definition 4.16 (Absolute value) If \(x\) is real number. We define the absolute value \(|x|\) of \(x\) is defined as follows. If \(x\) is positive,then \(|x|:=x\). If \(x\) is negative,then \(|x|:=-x\). If \(x\) is zero, then \(|x|:=0\).
Definition 4.17 (Ordering of the real numbers) Let \(x\) and \(y\) be real numbers. We say that \(x>y\) iff \(x-y\) is a positive rational number, and write \(x<y\) iff \(x-y\) is a negative rational number. We write \(x\geq y\) iff either \(x>y\) or \(x=y\), and similarly define \(x\leq y\).
Proposition 4.4 All the claims in Proposition 4.4 which held for rationals, continue to hold for real numbers.
Proposition 4.5 Let \(x\) be a positive real number. Then \(x^{-1}\) is also positive. Also, if \(y\) is another positive number and \(x>y\), then \(x^{-1} < y^{-1}\).
Proposition 4.6 (The non-negative reals are closed) Let \(a_1,a_2,\dots\) be a Cauchy sequence of non-negative rational numbers. Then \({\rm LIM}_{n\to \infty}a_n\) is a non-negative real number.
Eventually we will see a better explanation of this fact: the set of non-negative reals is closed, whereas the set of positive reals is open. See Section 10.
Corollary 4.1 Let \((a_n)_{n=1}^{\infty}\) and \((b_n)_{n=1}^{\infty}\) be Cauchy sequence of rationals such that \(a_n\geq b_n\) for all \(n\geq 1\). Then \({\rm LIM}_{n\to \infty}a_n \geq {\rm LIM}_{n\to \infty}b_n\).
We now define distance \(d(x,y):=|x-y|\) just as we did for the rationals. In fact Proposition 4.5 and 4.6 hold out not only for the rationals, but for the reals; the proof is identical.
Proposition 4.7 (Bounding of reals by rationals) Let \(x\) be a positive real number. Then there exists a positive rational number \(q\) such that \(q\leq x\), and there exists a positive integer \(N\) such that \(x\leq N\).
One can prove this with Proposition 4.8 and Corollary 4.1.
Corollary 4.2 (Archimedean property) Let \(x\) and \(\varepsilon\) be any positive real numbers. THen there exists a positive integer \(M\) such that \(M\varepsilon>x\).
Proposition 4.8 Given any two real numbers \(x<y\), we can find a rational number \(q\) such that \(x<q<y\).
Exercise 4.2 Show that for every real number \(x\) there is exactly one integer \(N\) such that \(N\leq x<N+1\).(This integer \(N\) is caled the integer part of \(x\), and is sometimes denoted \(N=[x]\).)
Exercise 4.3 Let \(x\) and \(y\) be real numbers. Show that \(x\leq y+\varepsilon\) for all numbers \(\varepsilon>0\) iff \(x\leq y\). Show that \(|x-y|\leq \varepsilon\) for all numbers \(\varepsilon>0\) iff \(x=y\).
4.5 The least upper bound property
Definition 4.18 (Upper bound) Let \(E\) be a subset of \(\mathbb{R}\), and let \(M\) be a real number. We say that \(M\) is an upper bound for \(E\), iff we have \(x\leq M\) for element \(x\) in \(E\).
Definition 4.19 (Least upper bound) Let \(E\) be a subset of \(\mathbb{R}\), and let \(M\) be a real number. We say that \(M\) is an least upper bound for \(E\), iff (a) \(M\) is an upper bound for \(E\), and also (b) any other upper bound \(M'\) for \(E\) numst be greater than or equal to \(M\).
Proposition 4.9 (Uniqueness of least upper bound) Let \(E\) be a subset of \(\mathbb{R}\). Then \(E\) can have at most one least upper bound.
Theorem 4.1 (Existence of least upper bound) Let \(E\) be a non-empty subset of \(\mathbb{R}\). If \(E\) has an upper bound, then it must have exactly one least bound.
Definition 4.20 (Supremum) Let \(E\) be subset of the real numbers. If \(E\) is non-empty and has some upper bound, we define \(\sup(E)\) to be the least upper bound of \(E\) (this is well-defined by Theorem 4.1). We introduce two dditional symbols, \(+\infty\) and \(-\infty\). If \(E\) is non-empty and has no upper bound, we set \(\sup(E):=+\infty\); if \(E\) is empty, we set \(\sup(E):=-\infty\). We refer to \(\sup(E)\) as the supremum of \(E\), and also denote it by \(\sup E\).
Proposition 4.10 There exists a positive real number \(x\) such that \(x^2=2\).
In Chapter 5 we will use the least lower bounds property to develop the theory of limits, which allows us to do many more things than just take square roots.
We can of course talk about lower bounds, and greatest lower bounds, of sets \(E\), which is also known as infimum of \(E\) and is denoted \(\inf(E)\) or \(\inf E\).
Exercise 4.4 Let \(E\) be a subset of \(\mathbb{R}\), and suppose that \(E\) has a least upper bound \(M\) which is a real number, i.e., \(M=\sup(E)\). Let \(-E\) be the set \[ -E:=\{-x:x\in E\} \] Show that \(-M\) is the greatest lower bound of \(-E\), i.e., \(-M=\inf(-E)\).
4.6 Real exponentiation, part I
Definition 4.21 (Exponentiation to a natural number) Let \(x\) be a real number. To raise \(x\) to the power \(0\), we define \(x^0:=1\); in particular we define \(0^0:=1\). Now suppose inductively that \(x^n\) has been defined for some natural number \(n\), then we define \(x^{n+1} : = x^n\times x\).
Definition 4.22 (Exponentiation to a negative number) Let \(x\) be a non-zero real number. Then for any negative integer \(-n\), we define \(x^{-n} := 1/x^n\).
Proposition 4.11 All the properties in Proposition 4.7 remain valid if \(x\) and \(y\) are assumed to be real numbers instead of rational numbers.
Definition 4.23 Let \(x\geq 0\) be a non-negative real, and let \(n\geq 1\) be a positive integer. We define \(x^{1/n}\), also known as the \(n^{\text{th}}\) root of \(x\), by the formula \[ x^{1/n}:=\sup\{y\in\mathbb{R}:y\geq0 \text{ and }y^n\leq x\}. \]
Lemma 4.9 (Existence of $n^{\text{th}}$ roots) Let \(x\geq 0\) be a non-negative real, an let \(n\geq 1\) be a positive integer. Then the set \(E:=\sup\{y\in\mathbb{R}:y\geq0 \text{ and }y^n\leq x\}\) is non-empty and is also bounded above. In particular \(x^{1/n}\) is a real number.
We list some basic properties of \(n^{\text{th}}\) root below.
Lemma 4.10 Let \(x,y\geq 0\) be non-negative reals, and let \(n,m\geq 1\) be positive integers.
- If \(y=x^{1/n}\), then \(y^n=x\).
- Conversely, if \(y^n=x\), then \(y=x^{1/n}\).
- \(x^{1/n}\) is a positive real number.
- We have \(x>y\) iff \(x^{1/n}>y^{1/n}\)
- If \(x>1\), then \(x^{1/k}\) is a decreasing function of \(k\). If \(x<1\), then \(x^{1/k}\) is an increasing function of \(k\). If \(x=1\), then \(x^{1/k}=1\) for all \(k\).
- We have \((xy)^{1/n}=x^{1/n}y^{1/n}\).
- We have \((x^{1/n})^{1/m}=x^{1/nm}\).
Now we define how to raise a positive real number \(x\) to a rational exponent \(q\).
Definition 4.24 Let \(x>0\) be a positive real number, and let \(q\) be a reational number. To define \(q=a/b\) for some integer \(a\) and positive integer \(b\), and define \[ x^q:=(x^{1/b})^a. \]
Lemma 4.11 Let \(a,a'\) be integers and \(b,b'\) be positive integers such that \(a/b=a'/b'\), and let \(x\) be a positive real number. Then we have \((x^{1/b'})^{a'} = (x^{1/b})^a\).
Some basic facts about rational exponentiation:
Lemma 4.12 Let \(x,y>0\) be positive reals, and let \(q,r\) be rationals.
- \(x^q\) is a positive real.
- \(x^{q+r} = x^q x^r\) and \((x^q)^r = x^{qr}\).
- \(x^{-q} = 1/x^q\)
- If \(q>0\), then \(x>y\) iff \(x^q>y^q\).
- If \(x>1\), then \(x^q>x^r\) iff \(q>r\). If \(x<1\), then \(x^q>x^r\) iff \(q<r\).