Chapter 1

Review of Riemann Integral

In this chapter, the definition and some basic results on the theory of Riemann integration are reviewed, and the limitations of Riemann integration are pointed out so as to convince the reader of the necessity for a more general integral.

1.1  Definition and Some Characterizations

Let f:[a,b]→R be a bounded function. The idea of a Riemann integral of f is to associate a unique number γ to f such that, in case f(x) ≥ 0 for all x ∈ [a, b], then γ can be thought of as the area of the region bounded by the graph of f, x-axis, and the lines with equations x = a and x = b. For its definition, first we consider a partition P of [a, b], that is, a finite set P: = {x_i:i = 0, 1, …, k} such that a = x₀ < x₁ < x₂ < … < x_k = b, usually written as

P:a=x0<x1<x2<⋯<xk=b,

and consider the sums

L(P,f):=∑i=1kmiΔxi,U(P,f):=∑i=1kMiΔxi,

where

mi=inf{f(x):xi−1≤x≤xi},Mi=sup{f(x):xi−1≤x≤xi}

and Δx_i = x_i − x_i−1 for i = 1, …, k. Clearly, for every partition P of [a, b],

L(P,f)≤U(P,f).

Note that if f(x) ≥ 0 for all x ∈ [a, b], then L(P, f) is the total area of the rectangles with side lengths m_i and x_i − x_i−1, and U(P, f) is the total area of the rectangles with side lengths M_i and widths x_i − x_i−1, for i = 1, …, k. Thus, it is intuitively clear that the required area, say γ, under the graph of f must satisfy the relation:

L(P,f)≤γ≤U(P,f)

for all partitions P of [a, b]. With this requirement in mind, we introduce the following definition.

Definition 1.1.1 A bounded function f:[a,b]→R is said to be Riemann integrable on [a, b] if there exists a unique γ∈R satisfying

L(P,f)≤γ≤U(P,f)

for all partitions P of [a, b]. If such a γ exists, then it is called the Riemann integral of f and it is denoted by

∫abf(x)dx.

We shall see that every bounded function f:[a,b]→R having at most a finite number of discontinuities is Riemann integrable. However, every bounded function f on [a, b] need not be Riemann integrable, as the following example shows.

Example 1.1.2 Let f:[0,1]→R be the Dirichlet function, that is,

f(x)={0ifxrational,1ifxirrational.

Note that, for any partition P of [a, b], we have L(P, f) = 0 and U(P, f) = 1. Thus, for every number α ∈ [0, 1], we have L(P, f) ≤ α ≤ U(P, f) for every partition P of [0, 1]. In other words, α∈R satisfying L(P, f) ≤ α ≤ U(P, f) for all partitions P is not unique. Hence, f is not Riemann integrable.

In the following, we shall denote the set of all partitions of [a, b] by P.

Let f:[a,b]→R be a bounded function and let

m=inf{f(x):a≤x≤b},M=sup{f(x):a≤x≤b}.

Then, for any partition P = {x_i: i = 0, 1, …, k} of [a, b], we have

L(P,f)=∑i=1kmiΔxi≥∑i=1kmΔxi=m(b−a)

and

U(P,f)=∑i=1kMiΔxi≤∑i=1kMΔxi=M(b−a)

so that

m(b−a)≤L(P,f)≤U(P,f)≤M(b−a).

Thus, the sets {L(P,f):P∈P} and {U(P,f):P∈P} are bounded above and bounded below. Hence,

L(f):=sup{L(P,f):P∈P},U(f):=inf{U(P,f):P∈P}

exist as real numbers. Using the quantities L(f) and U(f), we have the following characterization of Riemann integrability.

Theorem 1.1.3 A bounded function f:[a,b]→R is Riemann integrable on [a, b] if and only if L(f) = U(f).

Proof. Suppose f:[a,b]→R is Riemann integrable on [a, b], and let γ be the Riemann integral of f. Then, from the relations

L(P,f)≤γ≤U(P,f)∀P∈P	(*)

we have

L(f)≤γ≤U(f).

Consequently,

L(P,f)≤L(f)≤γ≤U(f)≤U(P,f)∀P∈P.

Now, since γ is the only number satisfying (*), we have L(f) = γ = U(f).

Conversely, suppose L(f) = U(f). Then we have

L(P,f)≤L(f)=U(f)≤U(P,f)∀P∈P.

Thus, γ: = L(f) = U(f) satisfies (*). Further, if γ~∈R satisfies

L(P,f)≤γ~≤U(P,f)∀P∈P,

then, from the definition of L(f) and U(f), it follows that L(f)≤γ~≤U(f). Hence, γ~=γ. Thus, we have proved that there exists a unique real number γ satisfying (*).▮

Remark 1.1.4 As you must have observed, the proof of Theorem 1.1.3 was very easy. We gave its proof in detail, mainly because of the fact that in standard text books, the Riemann integrability of a bounded function f:[a,b]→R is defined by requiring L(f) = U(f).♢

Remark 1.1.5 The quantities L(P, f) and U(P, f) are known as lower Darboux sum and upper Darboux sum, respectively. Analogously, the quantities L(f) and U(f) are known as lower Darboux integral and upper Darboux integral, respectively. Therefore, in view of Theorem 1.1.3, the integral ∫abf(x)dx in Definition 1.1.1 is also known as the Darboux-Riemann integral.♢

Let P be a partition of [a, b]. A new partition of [a, b] obtained from P by adjoining additional points is called a refinement of P. Thus, if P = {x_i: i = 1, …, k} is a partition of [a, b], P∪{t1,…,tℓ}, with each t_j satisfying x_i−1 < t_j < x_i for some i ∈ {1, …, k}, is a refinement of P.

If P₁ and P₂ are partitions of [a, b], we can consider a new partition P, which is a refinement of both P₁ and P₂, by using all the partition points of P₁ and P₂, taking repeated points only once. Such a partition is usually denoted by P1∪P2.

Given any two partitions P and Q of [a, b], it can be seen that

L(P,f)≤L(P∪Q,f)andU(P∪Q,f)≤U(Q,f).

Thus, L(P, f) ≤ U(Q, f) for any partitions P and Q of [a, b]. Consequently,

L(f)≤U(f).

Exercise 1.1.6 Prove the relations in (*) above.♢

The characterization given in the following theorem is useful in deducing many properties of Riemann integral.

Theorem 1.1.7 Let f:[a,b]→R be a bounded function. Then f is Riemann integrable if and only if for every ɛ > 0, there exists a partition P of [a, b] such that U(P, f) − L(P, f) < ɛ.

Proof. Suppose f is Riemann integrable and let ɛ > 0 be given. By the definitions of L(f) and U(f), there exist partitions P₁ and P₂ of [a, b] such that

L(f)−ε/2<L(P1,f)andU(P2,f)<U(f)+ε/2.

Let P=P1∪P2, the partition obtained by combining P₁ and P₂. Then, we have

L(f)−ε/2<L(P1,f)≤L(P,f)≤U(P,f)≤U(P2,f)<U(f)+ε/2.

Since L(f) = U(f) (cf. Theorem 1.1.3), it follows that

U(P,f)−L(P,f)<[U(f)+ε/2]−[L(f)−ε/2]=ε.

Conversely, suppose that for every ɛ > 0, there exists a partition P of [a, b] such that U(P, f) − L(P, f) < ɛ. Since

L(P,f)≤L(f)≤U(f)≤U(P,f),

we have

U(f)−L(f)≤U(P,f)−L(P,f)<ε.

This is true for every ɛ > 0. Hence, L(f) = U(f), and hence f is Riemann integrable.▮

Here is an immediate consequence of the above theorem.

Corollary 1.1.8 A bounded function f:[a,b]→R is Riemann integrable if and only if there exists a sequence (P_n) of partitions of [a, b] such that

U(Pn,f)−L(Pn,f)→0asn→∞,

and in that case the sequences (U(P_n, f)) and (L(P_n, f)) converge to the same limit ∫abf(x)dx.

Exercise 1.1.9 Give proof for the above corollary.♢

Next we give another characterization of Riemann integrability. For that purpose we introduce the following definition.

Definition 1.1.10 Let P: a = x₀ < x₁ < · · · < x_n = b be a partition of [a, b] and let T: = {t_i : i = 1, …, n} with t_i ∈ [x_i−1, x_i], i = 1, …, n. The set T is called a tag set for P. Given a function f:[a,b]→R, the sum

S(P,T,f):=∑i=1nf(ti)Δxi

is called the Riemann sum of f associated with (P, T). The quantity

|P|:=max{xi−xi−1:i=1,…,n}

is called the mesh of the partition P.♢

Note that the Riemann sums may vary as the tag sets vary. It is obvious that, if f:[a,b]→R is a bounded function, then

L(P,f)≤S(P,T,f)≤U(P,f)

for any partition P of [a, b] and for any tag set T for P. Therefore, by Theorem 1.1.7, we have the following result.

Theorem 1.1.11 If f:[a,b]→R is Riemann integrable, then for every ɛ > 0, there exists a partition P such that

|S(P,T,f)−∫abf(x)dx|<ε

for every tag set T for P.

Exercise 1.1.12 Supply details of the proof of the above theorem.♢

In fact, the converse to Theorem 1.1.11 is also true.

Theorem 1.1.13 Let f:[a,b]→R be a bounded function. Suppose there exists γ∈R such that for every ɛ > 0, there exists a partition P of [a, b] satisfying

|S(P,T,f)−γ|<ε

for every tag set T of P. Then f is Riemann integrable and γ=∫abf(x)dx.

Proof. Let ɛ > 0 be given and let P: a = x₀ < x₁ < · · · < x_k = b be as in the hypothesis of the theorem. Then we have

γ−ε<S(P,T,f)<γ+ε	(1)

for any tag set T corresponding to P. Let u_i, v_i ∈ [x_i−1, x_i] for i = 1, …, k be such that

Mi−ε<f(ui)andf(vi)<mi+εfori=1,…,k.	(2)

Consider the tag sets T₁ = {u_i: i = 1, …, k} and T₂ = {v_i: i = 1, …, k} for the partition P. Then from (2), we have

U(P,f)−ε(b−a)<S(P,T1,f),S(P,T2,f)<L(P,f)+ε(b−a).

This, together with (1), implies

U(P,f)−ε(b−a)<γ+ε,γ−ε<L(P,f)+ε(b−a).

In particular,

U(P,f)−γ<ε[(b−a)+1],γ−L(P,f)<ε[(b−a)+1]	(3)

so that

U(P,f)−L(P,f)<2ε[(b−a)+1].

Hence, by Theorem 1.1.7, f is Riemann integrable. Therefore, by the relations in (3), we have

γ−ε[(b−a)+1]<L(P,f)≤∫abf(x)dx≤U(P,f)<γ<ε[(b−a)+1].

Thus,

|γ−∫abf(x)dx|<ε[(b−a)+1]

for every ɛ > 0. Consequently, γ=∫abf(x)dx.▮

If we know that f is Riemann integrable, then the following theorem is better suited for obtaining approximations for ∫abf(x)dx. For its proof, one may refer Ghorpade and Limaye [7].

Theorem 1.1.14 Suppose f:[a,b]→R is a Riemann integrable function. Then for every ɛ > 0, there exists a δ > 0 such that

|S(P,T,f)−∫abf(x)dx|<ε

for any partition P of [a, b] with |P| < δ and for every tag set T for P.

The conclusion in the above theorem is usually written as

lim|P|→0S(P,T,f)=∫abf(x)dx.

Here is an immediate consequence of Theorem 1.1.14.

Corollary 1.1.15 Suppose f:[a,b]→R is a Riemann integrable function and (P_n) is a sequence of partitions of [a, b] such that |P_n| → 0 as n → ∞. If T_n is a tag set for P_n for each n∈N, then

S(Pn,Tn,f)→∫abf(x)dxasn→∞.

Looking at Theorem 1.1.13, one may ask the following question.

If (P_n) is a sequence of partitions of [a, b] such that limn→∞|Pn|=0 and limn→∞S(Pn,Tn,f)=γ for some γ∈R, where T_n is a tag set for P_n for each n∈N, then, is it true that f is Riemann integrable and γ=∫abf(x)dx?

The answer is in the negative as the following example shows.

Example 1.1.16 Consider the Dirichlet function f:[0,1]→R, of Example 1.1.2. That is,

f(x):={0,x∈Q,1,x∉Q.

Let x_i: = i/n for i = 0, 1, …, n, and let t_i be any rational point in the interval [x_i−1, x_i]. In this case we have limn→∞|Pn|=0 and S(P_n, T_n, f) = 0 for all n∈N so that limn→∞S(Pn,Tn,f)=0. However, f is not Riemann integrable.♢

1.2  Advantages and Some Disadvantages

We may observe, in view of Corollary 1.1.8, that if (P_n) is a sequence of partitions of [a, b] such that (U(P_n, f)) and (L(P_n, f)) converge to the same limit say γ, then f is Riemann integrable, and γ=∫abf(x)dx. If f:[a,b]→R is a continuous function, then using the uniform continuity of f, it can be shown that for any sequence (P_n) of partitions of [a, b] satisfying |Pn|→0asn→∞, we have U(P_n, f) − L(P_n, f) → 0 as n → ∞. Thus, by Corollary 1.1.8, we can conclude the following:

Every continuous function f:[a,b]→R is Riemann integrable.

The following results are also true; for their proofs, the reader may refer to Ghorpade and Limaye [7] or Rudin [13].

(a)Every bounded function f:[a,b]→R having at most a finite number of discontinuities is Riemann integrable.

(b)Every monotonic function f:[a,b]→R is Riemann integrable.

Thus, the set of all Riemann integrable functions is very large. In fact we have the following theorem, known as Lebesgue’s criterion for Riemann integrability, whose proof depends on some techniques involving the concept of oscillation of a function; refer to Delninger [4] for its proof.

Lebesgue’s criterion for Riemann integrability: A bounded function f:[a,b]→R is Riemann integrable if and only if the set of points at which f is discontinuous is of measure zero.

In the above, the terminology set of measure zero is used in the sense of the following definition.

Definition 1.2.1 A set E⊆R is said to be of measure zero if for every ɛ > 0, there exists a countable family {I_n} of open intervals such that

E⊆⋃nInand∑nℓ(In)<ε,

where ℓ(I_n) is the length of the interval I_n.♢

Example 1.2.2 We show that every countable subset of R is of measure zero. To see this, consider a countable set E = {a_n: n ∈ Λ}, where Λ is {1, …, k} for some k∈N or Λ=N. For ɛ > 0, let

In:=(an−ε/2n+1,an+ε/2n+1),n∈Λ.

Then

E⊆∪n∈ΛInand∑n∈Λℓ(In)=∑n∈Λ(ε/2n)≤ε.

♢

Can an uncountable set be of measure zero? We shall answer this question affirmatively in the next chapter.

Functions with only a finite or countably infinite number of discontinuities in [a, b] can be constructed easily. In fact, the following example shows that given any countable subset S of [a, b], there is a function f:[a,b]→R such that S is exactly the set of points in [a, b] at which f is discontinuous.

Example 1.2.3 Let I = [a, b], S:={an:n∈N}⊆I and let f:I→R be defined by f(a_n) = 1/n for all n∈N and f(x) = 0 for x∈I∖S. Clearly, this function is not continuous at any x ∈ S. We show that f is continuous at every x∈I∖S.

Let x0∈I∖S. Then f(x₀) = 0. For ɛ > 0, we have to find a δ > 0 such that |x − x₀| < δ implies |f(x)| < ɛ.

For δ > 0, let Jδ:=(x0−δ,x0+δ)∩I. For ɛ > 0, let k∈N be such that 1/k < ɛ. Choose δ > 0 such that

a1,a2,…,ak∉Jδ.

For instance, we may choose 0 < δ < min{|x₀ − a_i|: i = 1, …, k}. Then we have

Jδ∩{a1,a2,…}⊆{ak+1,ak+2,…}.

Hence, for x∈Jδ, we have either f(x) = 0 or f(x) = 1/n for some n > k. Thus,

|f(x)|≤1k<ε.

Thus we have proved that f is continuous at x₀.

If we take S as the set of all rational numbers in I, then the function f is continuous at every irrational number in I and discontinuous at every rational number in I.

Although the set of Riemann integrable functions on [a, b] is quite large, this class lacks some desirable properties. For example observe the following drawbacks of Riemann integrability and Riemann integration:

(a)If (f_n) is a sequence of Riemann integrable functions on [a, b] and if f_n(x) → f(x) as n → ∞ for every x ∈ [a, b], then it is not necessary that f is Riemann integrable.

(b)Even if the function f in (a) is Riemann integrable, it is not necessary that ∫abfn(x)dx→∫abf(x)dx as n → ∞.

To illustrate the last two statements consider the following examples.

Example 1.2.4 Let {r₁, r₂, …} be an enumeration of the set rational numbers in [0, 1]. For each n∈N, let

fn(x)={0ifx∈{r1,…,rn},1ifx∉{r1,…,rn}.

Then each f_n is Riemann integrable, as it is continuous except at a finite number of points. Note that f_n(x) → f(x) as n → ∞ for every x ∈ [0, 1], where f:[0,1]→R is the Dirichlet’s function, that is,

f(x)={0ifxrational,1ifxirrational.

We have seen in Example 1.1.2 that f is not Riemann integrable, which also follows from the Lebesgue’s criterion of Riemann integrability, since f is discontinuous everywhere. Thus, though ∫01fn(x)dx=1 for every n∈N and f_n(x) → f(x) for every x ∈ [0, 1], we cannot even talk about the integral of f.♢

Example 1.2.5 For n∈N, let fn:[0,1]→R be defined by

fn(x)=nχ(0,1/n](x),x∈[0,1],

where χ_E denotes the characteristic function of E, that is,

χE(x)={ 1ifx∈E,0ifx∈E.

Then we see that, for each x ∈ [0, 1], f_n(x) → f(x) = 0 as n → ∞, but ∫01fn(x)dx=1 for every n∈N. Hence ∫abfn(x)dx↛∫abf(x)dx.

Example 1.2.6 Consider fn:[0,1]→R defined by

fn(x)={ne−nxifx∈(0,1],0ifx=0.

Then we have, for each x ∈ [0, 1], f_n(x) → f(x) = 0 as n → ∞. Note that ∫01fn(x)dx=1−e−n→1 as n → ∞. Thus, ∫01fn(x)dx↛∫01f(x)dx.♢

In Examples 1.2.5 and 1.2.6, we see that, although the sequence (f_n(x)) converges for each x ∈ [0, 1], it is not uniformly bounded, that is, there does not exist an M > 0 such that |f_n(x)| ≤ M for all x ∈ [a, b] and for all n∈N. In fact, in both the examples, the sequence (f_n(1/n)) is unbounded. In this context, it is worth mentioning the following theorem, known as Arzela’s dominated convergence theorem, also called Arzela’s theorem.

Theorem 1.2.7 (Arzela’s theorem) Suppose (f_n) is a sequence of Riemann integrable functions defined on [a, b] such that f_n(x) → f(x) as n → ∞ for each x ∈ [a, b] for some Riemann integrable function f. If (f_n) is uniformly bounded, then

∫abfn(x)dx→∫abf(x)dxasn→∞.

For a recent elementary proof for the above theorem, one may refer to [8].

Note that, in Arzela’s theorem, we assumed that the limit function f is Riemann integrable. In this course we shall have a new type of integral, called the Lebesgue integral, which includes the Riemann integral, and derive Arzela’s theorem as a consequence of a more general result (see Theorem 5.1.12). In fact, the assumption of Riemann integrability of f is not required, but then, the integral of f is to be understood in the sense of the Lebesgue integral.

In the next section we introduce certain notations and conventions which we shall use throughout the book.

1.3  Notations and Conventions

By countable family {A_n} of sets, we mean a family {A_n: n ∈ Λ} of sets A_n, n ∈ Λ, where either Λ = {1, 2, …, k} for some k∈N or Λ=N, the set of natural numbers. Then, ∪{An:n∈Λ} and ∩{An:n∈Λ} will be written as ∪nAn and ∩nAn, respectively. Also, for a countable set {an∈R:n∈Λ} of non-negative real numbers, the series ∑n∈Λan will be written as ∑nan.

By an interval we mean a subset I of R which is any of the following forms (a, b), [a, b), (a, b], [a, b] for any a,b∈R with a < b or of the forms (a, ∞), [a, ∞), ( − ∞, a), ( − ∞, a] for any a∈R. Here, for a∈R,

(a,∞):={x∈R:a<x},

[a,∞):={x∈R:a≤x},

(−∞,a):={x∈R:x<a},

(−∞,a]:={x∈R:x≤a}.

The intervals (a, b), [a, b), (a, b], [a, b] are bounded intervals with end points a and b, and the intervals (a, ∞), [a, ∞), ( − ∞, a), ( − ∞, a], ( − ∞, ∞) are unbounded intervals.

Length of an interval I is denoted by ℓ(I). Thus, for a bounded interval I of end points a, b with a < b, ℓ(I) is b − a. If I is an unbounded interval, then we say that its length is infinity, and we write ℓ(I) = ∞. Also, for a divergent series ∑n∈Λan with a_n ≥ 0, we write ∑n∈Λan=∞.

Although ∞ is not a real number, we shall use the symbol ∞ as an extended real number.

We have already used the symbol ∞ to denote the sum of a divergent series of non-negative real numbers. Similarly, we may use the symbol −∞ for the sum of a divergent series of non-positive real numbers.

Thus, we have the set R~:=R∪{∞,−∞}, called the set of all extended real numbers. The order relations, addition and multiplication on R~ are defined as follows: For every a∈R,

∞>a,−∞<a,±∞±a=±∞,

±∞×a={±∞ if a>0,∓∞ if a<0.

Further,

±∞×0=0,±∞±∞=±∞,(±∞)×(±∞)=∞.

However, ±∞∓∞ is not defined.

In the above, addition and multiplication are commutative. Also, for any a∈R, we denote

(a,∞]:=(a,∞)∪{∞},[−∞,a):=(−∞,a)∪{−∞},

[a,∞]:=[a,∞)∪{∞},[−∞,a]:=(−∞,a]∪{−∞}.

We shall also use the notation [ − ∞, ∞] for the set R~=R∪{−∞,∞}, the set of all extended real numbers. Intervals of the form (a, ∞] and [ − ∞, b) for a,b∈R are called neighbourhoods of ∞ and −∞, respectively.

Throughout the text, we consider R with the usual metric, that is,

d(x,y):=|x−y|,x,y∈R.

Thus, a set A⊆R is open in R if and only if for each x ∈ A, there exists r > 0 such that (x − r, x + r)⊆ A, and a set B⊆R is closed if and only if its complement is open in R, that is the set R∖B, is open. Further concepts based on open subsets of R will be introduced as and when they are required.

Let S⊆R. Then

(a)S is said to be bounded above if there exists b∈R such that s ≤ b for all s ∈ S, and in that case, b is called an upper bound of S;

(b)S is said to be bounded below if there exists a∈R such that a ≤ s for all s ∈ S, and in that case, a is called a lower bound of S.

(c)S is said to have a least upper bound b0∈R if b₀ is an upper bound of S and for any b < b₀, there exists s ∈ S such that b < s ≤ b₀.

(d)S is said to have a greatest lower bound a0∈R if a₀ is a lower bound of S and for any a > a₀, there exists s ∈ S such that a₀ ≤ s < a.

It can be easily seen that if S⊆R has a least upper bound (respectively, greatest lower bound), then it is unique. An important property of R is its least upper bound property, stated as follows:

Every subset of R which is bounded above has the least upper bound.

Using the least upper bound property of R, it can be deduced that it has greatest lower bound property as well:

Every subset of R which is bounded below has the greatest lower bound.

If S⊆R is bounded above, then its least upper bound is also called the supremum of S, and it is denoted by sup (S). If S⊆R is bounded below, then its greatest lower bound is also called the infimum of S, and it is denoted by inf (S).

If S is not bounded above, then we shall write sup (S) = ∞, and if S is not bounded below, then we shall write inf (S) = −∞. This convention is also adopted if S is a subset of [ − ∞, ∞].

Using the above definitions, supremum and infimum of a sequence in R~ can be defined as follows: Let (a_n) be a sequence in R~. Then we define

supnan=sup{an:n∈N},infnan=inf{an:n∈N}.

Also, the limit supremum and limit infimum of (a_n) are defined by

lim supnan=infksupn≥kan,lim infnan=supkinfn≥kan,

respectively. We observe that, if

bk:=supn≥kan,ck:=infn≥kan

for k∈N, then (b_k) is a decreasing sequence and (c_k) is an increasing sequence, and hence, they converge in R~. Thus, we have

lim supnan=limk→∞supn≥kan,lim infnan=limk→∞infn≥kan.

For a sequence (a_n) in R~, if lim supnan=lim infnan, then we say that (a_n) converges in R~, and the common value, say a∈R~, is called the limit of (a_n), and in that case we write

limn→∞an=aoran→a as n→∞oran→a.

Let (a_n) be a sequence in R~. Then the following statements can be easily verified.

1. (a_n) converges in R~ to a∈R if and only if for every ɛ > 0, there exists N∈N such that a_n ∈ (a − ɛ, a + ɛ) for all n ≥ N.
2. (a_n) converges in R~ to ∞ if and only if for every r∈R, there exists N∈N such that a_n ∈ (r, ∞] for all n ≥ N.
3. (a_n) converges in R~ to −∞ if and only if for every r∈R, there exists N∈N such that a_n ∈ [ − ∞, r) for all n ≥ N.
4. If an∈R for all n∈N and (a_n) converges in R, then (a_n) converges in R~.
5. If an∈R for all n∈N and (a_n) diverges to ∞ (respectively, to −∞), then it converges in R~ to ∞ (respectively, to −∞).