Kenneth Kuttler

13.3. CONVERGENT SEQUENCES, SEQUENTIAL COMPACTNESS 239

Lemma 13.2.3 Let Ik = ∏pi=1

[ak

i ,bki]≡{x ∈ Rp : xi ∈

[ak

i ,bki]}

and suppose that for allk = 1,2, · · · ,

Ik ⊇ Ik+1.

Then there exists a point c ∈ Rp which is an element of every Ik. If limk→∞ diam(Ik) = 0,then the point c is unique.

Proof: For each i = 1, · · · , p,[ak

i ,bki]⊇[ak+1

i ,bk+1i

]and so, by Lemma 13.2.1, there

exists a point ci ∈[ak

i ,bki]

for all k. Then letting c≡ (c1, · · · ,cp) it follows c ∈ Ik for all k.If the condition on the diameters holds, then the lengths of the intervals limk→∞

[ak

i ,bki]= 0

and so by the same lemma, each ci is unique. Hence c is unique. ■I will sometimes refer to the above Cartesian product of closed intervals as an interval

to emphasize the analogy with one dimensions, and sometimes as a box.

13.3 Convergent Sequences, Sequential CompactnessA mapping f : {k,k+1,k+2, · · ·}→ Rp is called a sequence. We usually write it in theform

{a j}

where it is understood that a j ≡ f ( j). In the same way as for sequences of realnumbers, one can define what it means for convergence to take place.

Definition 13.3.1 A sequence, {ak} is said to converge to a if for every ε > 0 there existsnε such that if n > nε , then |a−an| < ε . The usual notation for this is limn→∞an = aalthough it is often written as an→ a.

One can also define a subsequence in the same way as in the case of real valued se-quences.

Definition 13.3.2{ank

}is a subsequence of {an} if n1 < n2 < · · · .

The following theorem says the limit, if it exists, is unique.

Theorem 13.3.3 If a sequence, {an} converges to a and to b then a= b.

Proof: There exists nε such that if n > nε then |an−a| < ε

2 and if n > nε , then|an−b|< ε

2 . Then pick such an n.

|a−b|< |a−an|+ |an−b|< ε

2= ε.

Since ε is arbitrary, this proves the theorem. ■The following is the definition of a Cauchy sequence in Rp.

Definition 13.3.4 {an} is a Cauchy sequence if for all ε > 0, there exists nε such thatwhenever n,m≥ nε ,

|an−am|< ε.

A sequence is Cauchy, means the terms are “bunching up to each other” as m,n getlarge.

Theorem 13.3.5 The set of terms in a Cauchy sequence in Rp is bounded in the sense thatfor all n, |an|< M for some M < ∞.

13.3. CONVERGENT SEQUENCES, SEQUENTIAL COMPACTNESS 239Lemma 13.2.3 Let i, = My [ak bt] = {x ER? ix € [ak bk] } and suppose that for allk=1,2,---,Ty 2D Wg.Then there exists a point c € R? which is an element of every I,. If limg_,.. diam (I) = 0,then the point c is unique.Proof: For each i = 1,--- , p, [ak bf] ») [ap ! bet] and so, by Lemma 13.2.1, thereexists a point cj € [ak bE] for all k. Then letting c = (c1,--- ,cp) it follows c € i, for all k.If the condition on the diameters holds, then the lengths of the intervals lim;_,.. lat, bk] =0and so by the same lemma, each c; is unique. Hence c is unique. MfI will sometimes refer to the above Cartesian product of closed intervals as an intervalto emphasize the analogy with one dimensions, and sometimes as a box.13.3 Convergent Sequences, Sequential CompactnessA mapping f : {k,k+1,k+2,---}—> R? is called a sequence. We usually write it in theform {a i} where it is understood that a; = f (/). In the same way as for sequences of realnumbers, one can define what it means for convergence to take place.Definition 13.3.1 A sequence, {a;} is said to converge to a if for every € > 0 there existsne such that ifn > ne, then |a—ay,| < €. The usual notation for this is limy-s0An = aalthough it is often written as Qn — a.One can also define a subsequence in the same way as in the case of real valued se-quences.Definition 13.3.2. {a,, } is a subsequence of {an} ifn, <2 <-+-.The following theorem says the limit, if it exists, is unique.Theorem 13.3.3 [fa sequence, {a,} converges to a and to b then a = b.Proof: There exists ng such that if n > ng then |a,—a| < 5 and if n > ne, then|a, — b| < 5. Then pick such an n.E=€.2E|a—b| < |a—a,|+]a, —b] < 5tSince € is arbitrary, this proves the theorem.The following is the definition of a Cauchy sequence in R?’.Definition 13.3.4 {a@,} is a Cauchy sequence if for all € > 0, there exists ng such thatwhenever n,m > neg,|Qn—Am| < €.A sequence is Cauchy, means the terms are “bunching up to each other” as m,n getlarge.Theorem 13.3.5 The set of terms in a Cauchy sequence in R? is bounded in the sense thatfor all n, |ay| <M for some M < ~.