62 CHAPTER 4. FUNCTIONS AND SEQUENCES
Proof: I will show the second claim because it includes the first as a special case.Letting ε > 0 be given, for all n large enough, |y− yn| < ε so y ≥ yn− ε. Similarly, for nlarge enough, x≤ xn + ε . Therefore,
y− x≥ yn− ε− (xn + ε)≥ (yn− xn)−2ε ≥−2ε
Since ε is arbitrary, it follows that y− x≥ 0.Another important observation is that if a sequence converges, then it must be bounded.
Proposition 4.4.14 Suppose xn→ x. Then ∥xn∥ is bounded by some M < ∞.
Proof: There exists N such that if n≥N, then ∥x− xn∥< 1. It follows from the triangleinequality, see Proposition 4.4.2, that for n ≥ N,∥xn∥ ≤ 1+ ∥x∥. There are only finitelymany xk for k < N and so for all k,
∥xk∥ ≤max{1+∥x∥ ,∥xk∥ : k ≤ N} ≡M < ∞.
4.5 Cauchy SequencesA Cauchy sequence is one which “bunches up”. This concept was developed by Bolzanoand Cauchy. It is a fundamental idea in analysis.
Definition 4.5.1 {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 4.5.2 The set of terms (values) of a Cauchy sequence in Fp is bounded.
Proof: Let ε = 1 in the definition of a Cauchy sequence and let n > n1. Then from thedefinition, ∥an−an1∥< 1. It follows from the triangle inequality that for all n > n1,∥an∥<1+∥an1∥ .Therefore, for all n,∥an∥ ≤ 1+∥an1∥+∑
n1k=1 ∥ak∥ .
Theorem 4.5.3 If a sequence {an} in Fp converges, then the sequence is a Cauchysequence.
Proof: Let ε > 0 be given and suppose an → a. Then from the definition of con-vergence, there exists nε such that if n > nε , it follows that ∥an−a∥ < ε
2 . Therefore, ifm,n≥ nε +1, it follows that ∥an−am∥ ≤ ∥an−a∥+∥a−am∥< ε
2 +ε
2 = ε showing that,since ε > 0 is arbitrary, {an} is a Cauchy sequence.
The following theorem is very useful and is likely the most important property ofCauchy sequences. You know that if a sequence converges, then every subsequence con-verges to the same thing. However, you can have a sequence which does not converge,an = (−1)n for example which has a convergent subsequence, nk = 2k in this example.This won’t happen with a Cauchy sequence.
Theorem 4.5.4 Suppose {an} is a Cauchy sequence in Fp and there exists a subse-quence,
{ank
}which converges to a. Then {an} also converges to a.
Proof: Let ε > 0 be given. There exists N such that if m,n > N, then ∥am−an∥< ε/2.Also there exists K such that if k > K, then
∥∥a−ank
∥∥ < ε/2. Then let k > max(K,N) .Then for such k,∥ak−a∥ ≤
∥∥ak−ank
∥∥+∥∥ank −a∥∥< ε/2+ ε/2 = ε.
This theorem holds in all instances where it makes sense to speak of Cauchy sequences.
62 CHAPTER 4. FUNCTIONS AND SEQUENCESProof: I will show the second claim because it includes the first as a special case.Letting € > 0 be given, for all n large enough, |y—y,| < € so y > y, — €. Similarly, for nlarge enough, x < x, + €. Therefore,y—xX>Yn—E— (Xp +E) = (Yn —Xn) — 2E > —2ESince € is arbitrary, it follows thaty—x>0. JAnother important observation is that if a sequence converges, then it must be bounded.Proposition 4.4.14 Suppose x, — x. Then ||x;|| is bounded by some M < ~.Proof: There exists N such that if n > N, then ||x —x,|| < 1. It follows from the triangleinequality, see Proposition 4.4.2, that for n > N,||x,|| < 1+ |lx||. There are only finitelymany x; for k < N and so for all k,||xq|| < max {1+ |x|], |lax|| kK <N}SM<o. OE4.5 Cauchy SequencesA Cauchy sequence is one which “bunches up”. This concept was developed by Bolzanoand Cauchy. It is a fundamental idea in analysis.Definition 4.5.1 {a,} is a Cauchy sequence if for all € > 0, there exists ng such thatwhenever n,m > Neg, |dn — Am| < €.A sequence is Cauchy means the terms are “bunching up to each other” as m,n getlarge.Theorem 4.5.2 The set of terms (values) of a Cauchy sequence in F? is bounded.Proof: Let € = | in the definition of a Cauchy sequence and let n > n,. Then from thedefinition, ||ay — dn, || < 1. It follows from the triangle inequality that for all n > 11, ||an|| <1+ ||an, || Therefore, for all n, |[an|] < 1+ |lan, || + Leb, |laell-Theorem 4.5.3 If a sequence {a,} in F? converges, then the sequence is a Cauchysequence.Proof: Let € > 0 be given and suppose a, — a. Then from the definition of con-vergence, there exists ng such that if n > ne, it follows that ||a, —al|| < 5. Therefore, ifm,n > ne +1, it follows that ||an —am|| < ||@n — || + ||a—am|| < § + § = € showing that,since € > 0 is arbitrary, {a,} is a Cauchy sequence.The following theorem is very useful and is likely the most important property ofCauchy sequences. You know that if a sequence converges, then every subsequence con-verges to the same thing. However, you can have a sequence which does not converge,Gy = (—1)" for example which has a convergent subsequence, nz = 2k in this example.This won’t happen with a Cauchy sequence.Theorem 4.5.4 Suppose {an} is a Cauchy sequence in F? and there exists a subse-quence, {an, } which converges to a. Then {a,} also converges to a.Proof: Let € > 0 be given. There exists N such that if m,n > N, then ||am_—ay|| < €/2.Also there exists K such that if k > K, then \|]a — an, | < €/2. Then let k > max(K,N).Then for such k,|}ax — al < |[ax — an, || + ||an, — al] < €/2+e/2=€. OfThis theorem holds in all instances where it makes sense to speak of Cauchy sequences.
