Kenneth Kuttler

248 CHAPTER 10. NORMED LINEAR SPACES

10.1.2 Cauchy Sequences, Completeness

Of course it does not go the other way. For example, you could let xn = (−1)n and it has aconvergent subsequence but fails to converge. Here d (x,y) = |x− y| and the metric spaceis just R.

However, there is a kind of sequence for which it does go the other way. This is calleda Cauchy sequence.

Definition 10.1.15 {xn} is called a Cauchy sequence if for every ε > 0 there exists N suchthat if m,n≥ N, then d (xn,xm)< ε .

Now the major theorem about this is the following.

Theorem 10.1.16 Let {xn} be a Cauchy sequence. Then it converges if and only if anysubsequence converges.

Proof: =⇒ This was just done above.⇐= Suppose now that {xn} is a Cauchy sequence and limk→∞ xnk = x. Then there exists

N1 such that if k > N1, then d(xnk ,x

)< ε/2. From the definition of what it means to be

Cauchy, there exists N2 such that if m,n≥N2, then d (xm,xn)< ε/2. Let N ≥max(N1,N2).Then if k ≥ N, then nk ≥ N and so

d (x,xk)≤ d(x,xnk

)+d(xnk ,xk

2= ε (10.1)

It follows from the definition that limk→∞ xk = x. ■

Definition 10.1.17 A metric space is said to be complete if every Cauchy sequence con-verges.

Another nice thing to note is this.

Proposition 10.1.18 If {xn} is a sequence and if p is a limit point of the set S = ∪∞n=1 {xn}

then there is a subsequence{

xnk

}such that limk→∞ xnk = x.

Proof: By Theorem 10.1.7, there exists a sequence of distinct points of S denoted as{yk} such that none of them equal p and limk→∞ yk = p. Thus B(p,r) contains infinitelymany different points of the set D, this for every r. Let xn1 ∈ B(p,1) where n1 is the firstindex such that xn1 ∈ B(p,1). Suppose xn1 , · · · ,xnk have been chosen, the ni increasing andlet 1 > δ 1 > δ 2 > · · ·> δ k where xni ∈ B(p,δ i) . Then let

δ k+1 ≤min{

12k+1 ,d

(p,xn j

),δ j, j = 1,2 · · · ,k

}Let xnk+1 ∈ B(p,δ k+1) where nk+1 is the first index such that xnk+1 is contained B(p,δ k+1).Then limk→∞ xnk = p. ■

Another useful result is the following.

Lemma 10.1.19 Suppose xn→ x and yn→ y. Then d (xn,yn)→ d (x,y).

248 CHAPTER 10. NORMED LINEAR SPACES10.1.2 Cauchy Sequences, CompletenessOf course it does not go the other way. For example, you could let x, = (—1)" and it has aconvergent subsequence but fails to converge. Here d (x,y) = |x—y| and the metric spaceis just R.However, there is a kind of sequence for which it does go the other way. This is calleda Cauchy sequence.Definition 10.1.15 {x,} is called a Cauchy sequence if for every € > 0 there exists N suchthat if m,n > N, then d(Xn,Xm) < €.Now the major theorem about this is the following.Theorem 10.1.16 Let {x,} be a Cauchy sequence. Then it converges if and only if anysubsequence converges.Proof: => This was just done above.<= Suppose now that {x,} is a Cauchy sequence and lim;_,..Xn, =x. Then there existsNj such that if k > Nj, then d (xp,,x) < €/2. From the definition of what it means to beCauchy, there exists Np such that if m,n > No, then d (%m,Xn) < €/2. Let N > max (Nj, N2).Then if k > N, then nj, > N and sod(x,x) <d (x,Xn,) +d (Xnj+Xk) < =+ Ex= (10.1)It follows from the definition that lim;_,..x, =x.Definition 10.1.17 A metric space is said to be complete if every Cauchy sequence con-verges.Another nice thing to note is this.Proposition 10.1.18 If {x,} is a sequence and if p is a limit point of the set S = U?_, {xn}then there is a subsequence {xn, } such that limg..Xn, = X.Proof: By Theorem 10.1.7, there exists a sequence of distinct points of S denoted as{y,} such that none of them equal p and limy_,.0y, = p. Thus B(p,r) contains infinitelymany different points of the set D, this for every r. Let x,, € B(p,1) where n, is the firstindex such that x,, € B (p, 1). Suppose Xn,3°** +Xn, have been chosen, the n; increasing andlet 1 > 6; > 62 >--- > dy where x, € B(p,6;). Then letfiOx41 < min | ep (Pot) Bind = 1,2: asLet xn,,, € B(p, 6x41) where nz is the first index such that x,,,, is contained B (p, x41).Then limy_,..Xn, = p.Another useful result is the following.Lemma 10.1.19 Suppose x, — x and y, > y. Then d (Xn, Yn) > d (x,y).