Chapter 4 The real numbers

4.1 Cauchy sequences

Definition 4.1 (Sequences) Let m be an integer. A sequence (an)n=m of rational numbers is ant function from the set {nZ:nm to Q, i.e., a mapping which assigns to each integer n greater than or equal to m, a rational number an. More informally, a sequence (an)n=m of rational numbers is a collection of rationals am,am+1,am+2,

Definition 4.2 ($\varepsilon$-steadness) Let ε>0. A sequence (an)n=0 is said to be ε-steady iff each pair aj,ak of sequence elements is ε-close for every natural number j,k. In order words, the sequence a0,a1, is ε-steady iff d(aj,ak)ε for all j,k.

Definition 4.3 (Eventual $\varepsilon$-steadness) Let ε>0. A sequence (an)n=0 is said to be eventually ε-steady iff the sequence aN,aN+1,aN+2, is ε-steady for some natural number N0. In other words, the sequence a0,a1, is eventually ε-steady iff there exists an N0 such that d(aj,ak)ε for all j,kN.

Definition 4.4 (Cauchy sequences) A sequence (an)n=0 of rational numbers is said to be Cauchy sequences iff for every rational ε>0, thesequence (an)n=0 is eventually ε-steady.

Definition 4.5 (Bounded sequences) Let M0 be rational. A finite sequence a1,a2,,an is bounded by M iff |ai|M for all 1in. An infinite sequence (an)n=1 is bounded by M iff iff |ai|M for all i1. 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 (an)n=1 is bounded.

4.2 Equivalent Cauchy sequences

Definition 4.6 ($\varepsilon$-close sequences) Let (an)n=0 and (bn)n=0 be two sequences, and let ε>0. We say the sequence (an)n=0 is ε-close to (bn)n=0 iff an is ε-close to bn for each nN. In other wors, the sequence (an)n=0 is ε-close to (bn)n=0 iff |anbn|ε for all n=0,1,2,.

Definition 4.7 ($Eventually \varepsilon$-close sequences) Let (an)n=0 and (bn)n=0 be two sequences, and let ε>0. We say the sequence (an)n=0 is eventually ε-close to (bn)n=0 iff there exists an N>0 such that the sequences (an)n=N and (bn)n=N are ε-close.

Definition 4.8 (Equivalent sequences) Two sequences (an)n=0 and (bn)n=0 are equivalent iff for each rational ε>0, the sequences (an)n=0 and (bn)n=0 are eventually ε-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 LIMnan, where (an)n=1 is a Cauchy sequence of rational numbers. Two real numbers LIMnan and LIMnbn are said to be equal iff (an)n=1 and (bn)n=1 are equivalent Cauchy sequences. The set of all real numbers is denoted R.

From now on we will refer to LIMnan as the formal limit of the sequence (an)n=1.

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=LIMnan and y=LIMnbn be real numbers. Then we define the sum x+y to be x+y:LIMn(an+bn).

Lemma 4.3 Let x=LIMnan and y=LIMnbn be real numbers. Then x+y is also a real number.

Lemma 4.4 Let x=LIMnan, y=LIMnbn and x=LIMnan 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=LIMnan and y=LIMnbn be real numbers. Then we define the product xy to be xy:LIMnanbn.

Lemma 4.5 Let x=LIMnan, y=LIMnbn and x=LIMnan be real numbers. The xy is also a real number. Furthermore, if x=x, then xy=xy.

At this point we embed the rationals back into the reals, by equating every rational number q with the real number LIMnq.

We can now easily define negation x for real numbers x by the formula x:=(1)×x, since 1 is a rational number and is hence real. Also, from our define it is clear that LIMnan=LIMn(an)

Once we have addition and negation, we can define substraction as usual by xy:=x+(y) note that this implies LIMnanLIMnbn=LIMn(anbn)

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 (an)n=1 of rational numbers is said to bounded away from zero iff there exists a rational numbers c>0 such that |an|>c for all n1.

Lemma 4.6 Let x be a non-zero real number. Then x=LIMnan for some Cauchy sequence (an)n=1 which is bounded away from zero.

Lemma 4.7 Suppose that (an)n=1 is a Cauchy sequence which is bounded away from zero. Then the sequence (a1n)n=1 is also a Cauchy sequence.

Definition 4.13 (Reciprocals of real numbers) Let x be a non-zero real number. Let (an)n=1 be a Cauchy sequence bounded away from zero such that x=LIMnan. Then we define the reciprocal x1 by the formula x1:=LIMna1n

Lemma 4.8 (Reciprocal is well-defined) Let (an)n=1 and (bn)n=1 be two Cauchy Sequences bounded away from zero such that LIMnan=LIMnbn. Then LIMna1n=LIMnb1n.

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×y1


Exercise 4.1 Let a,b be rational numbers. Show that a=b iff LIMna=LIMnb. 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 (an)n=1 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 anc for all n1. The sequence is negative bounded away from zero iff we have positive rational c>0 such that anc for all n1.

Definition 4.15 A real number x is said to be positive iff it can be written as x=LIMnan for some Cauchy sequence (an)n=1 which is positively bounded away from zero. x is said to be negative iff it can be written as x=LIMnan for some Cauchy sequence (an)n=1 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 xy is a positive rational number, and write x<y iff xy is a negative rational number. We write xy iff either x>y or x=y, and similarly define xy.

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 x1 is also positive. Also, if y is another positive number and x>y, then x1<y1.

Proposition 4.6 (The non-negative reals are closed) Let a1,a2, be a Cauchy sequence of non-negative rational numbers. Then LIMnan 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 (an)n=1 and (bn)n=1 be Cauchy sequence of rationals such that anbn for all n1. Then LIMnanLIMnbn.

We now define distance d(x,y):=|xy| 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 qx, and there exists a positive integer N such that xN.

One can prove this with Proposition 4.8 and Corollary 4.1.

Corollary 4.2 (Archimedean property) Let x and ε be any positive real numbers. THen there exists a positive integer M such that Mε>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 Nx<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 xy+ε for all numbers ε>0 iff xy. Show that |xy|ε for all numbers ε>0 iff x=y.

4.5 The least upper bound property

Definition 4.18 (Upper bound) Let E be a subset of R, and let M be a real number. We say that M is an upper bound for E, iff we have xM for element x in E.

Definition 4.19 (Least upper bound) Let E be a subset of 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 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 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, + and . If E is non-empty and has no upper bound, we set sup(E):=+; if E is empty, we set sup(E):=. We refer to sup(E) as the supremum of E, and also denote it by supE.

Proposition 4.10 There exists a positive real number x such that x2=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 infE.


Exercise 4.4 Let E be a subset of 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:xE} 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 x0:=1; in particular we define 00:=1. Now suppose inductively that xn has been defined for some natural number n, then we define xn+1:=xn×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 xn:=1/xn.

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 x0 be a non-negative real, and let n1 be a positive integer. We define x1/n, also known as the nth root of x, by the formula x1/n:=sup{yR:y0 and ynx}.

Lemma 4.9 (Existence of $n^{\text{th}}$ roots) Let x0 be a non-negative real, an let n1 be a positive integer. Then the set E:=sup{yR:y0 and ynx} is non-empty and is also bounded above. In particular x1/n is a real number.

We list some basic properties of nth root below.

Lemma 4.10 Let x,y0 be non-negative reals, and let n,m1 be positive integers.

  • If y=x1/n, then yn=x.
  • Conversely, if yn=x, then y=x1/n.
  • x1/n is a positive real number.
  • We have x>y iff x1/n>y1/n
  • If x>1, then x1/k is a decreasing function of k. If x<1, then x1/k is an increasing function of k. If x=1, then x1/k=1 for all k.
  • We have (xy)1/n=x1/ny1/n.
  • We have (x1/n)1/m=x1/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 xq:=(x1/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 (x1/b)a=(x1/b)a.

Some basic facts about rational exponentiation:

Lemma 4.12 Let x,y>0 be positive reals, and let q,r be rationals.

  • xq is a positive real.
  • xq+r=xqxr and (xq)r=xqr.
  • xq=1/xq
  • If q>0, then x>y iff xq>yq.
  • If x>1, then xq>xr iff q>r. If x<1, then xq>xr iff q<r.

4.7 Foot Notes

The system N, Q, and R stand for “natural”, “quotient”, and “real” respectively. Z stands for “Zahlen”, the German word for number.