In this section we establish a theorem that is useful in the further development of differentiation and integration. The Heine–Borel theorem shows that “covering” a set with a collection of special sets is particularly helpful in verifying uniform continuity. Consider, for example, the collection of intervals(1.9)

Any collection of intervals such as(1.9) is called a family of intervals. The symbol F is usually used to denote the totality of intervals in the family. Each interval is called a member (or element) of the family. Care must be exercised to distinguish the points of a particular interval from the interval itself. In the above example, the interval J ={x :1/3 <x< 2/3 } is not a member of F. However, every point of J is in every interval In for n> 2. The points0 <x< 1 are not themselves members of F although each point of 0 <x< 1 is in at least one I_n.

Definitions

A family F of intervals is said to cover the set S in R1 if each point of S is in at least one of the intervals of F. A family F₁ is a subfamily of F if each member of F₁ is a member of F.

For example, the intervals(1.10)

form a subfamily of family (1.9). Family (1.9) covers the set S ={x :0 < x< 1}, since every number x such that 0 <x< 1isin In for all n larger than 1/x. Thus every x is in at least one In. Family (1.10) also covers S.

More generally, we may consider a collection of sets A₁,A₂ ...A_n,..., which we call a family of sets. We use the same symbol F to denote such a family. A family F covers a set S if and only if every point of S is a point in at least one member of F. The family F,

Covers all of R¹. It is simple to verify that if any interval is removed from F, the resulting family fails to cover R1. For example, if J₁ is removed, then no member of the remaining intervals of F contains the number 2.

We shall be interested in families of open intervals that contain infinitely many members and that cover a set S. We shall examine those sets S that have the property that they are covered by a finite subfamily, i.e., a subfamily with only a finite number of members. For example, the family F of intervals K_n ={x :1/(n + 2)<x< 1/n}, n = 1, 2,...,

Covers the set S {x :0 <x< 1/2 }, as is easily verified. It may also be verified that no finite subfamily of F ={K_n} covers S. However, if we consider the set S₁ ={x : ε<x< 1 2 } for any ε> 0, it is clear that S₁ is covered by the finite subfamily {K₁,K₂,...,K_n} for any n such that n + 2 > 1/ε.

We may define families of interval sindirectly. For example, suppose that f : x → 1/x with domain D ={x :0 <x ≤ 1} is given. We obtain a family F of intervals by considering all solution sets of the inequality(1.11)

for every a ∈ D. For each value a, the interval(1.12)

Is the solution of Inequality (1.11). This family F ={La} covers the set D ={x :0 <x ≤ 1}, but no finite subfamily covers D.

Theorem 1.1 (Heine–Borel theorem)

Suppose that a family F of open intervals covers the closed interval I ={x:a ≤ x ≤ b}. Then a finite subfamily of F covers I.

Proof

We shall suppose that an infinite number of members of F are required to cover I and reach a contradiction. Divide I into two equal parts at the midpoint. Then an infinite number of members of F are required to cover either the left subinterval or the right subinterval of I. Denote by I₁ ={x : a₁ ≤ x ≤ b₁} the particular subinterval needing this infinity of members of F. We proceed by dividing I₁ into two equal parts, and denote by I₂ ={x : a₂ ≤ x ≤ b₂} that half of I1 that requires an infinite number of members of F. Repeating the argument, we obtain a sequence of closed intervals I_n ={x : a_n ≤ x ≤ b_n}, n = 1, 2,..., each of which requires an infinite number of intervals of F in order to be covered. Since b_n − a_n= (b − a)/2ⁿ → 0as n →∞, the Nested intervals theorem states that there is a unique number x₀ ∈ I that is in every In. However, since F covers I, there is a member of F, say J ={x : α<x<β}, such that x₀ ∈ J . By choosing N sufficiently large, we can find an IN contained in J . But this contradicts the fact that infinitely many intervals of F are required to cover IN, since we found that the one interval J covers IN.

In the Heine–Borel theorem, the hypothesis that I ={x : a ≤ x ≤ b} Is a closed interval iscrucial. The open interval J₁ ={x :0 <x< 1} is covered by the family F of intervals(1.9), but no finite subfamily of F covers J₁.

Problems

In each of Problems 1 through 6 decide whether or not the infinite sequence is a Cauchy sequence. If it is not a Cauchy sequence, find at least one subsequence that is a Cauchy sequence. In each case n = 1, 2,....

Basic Elements of Real Analysis, Murray H. Protter, Springer, 1998 .Page(63-65)