University:
The University Of Western Ontario
Course:
MATH 1000A | Introduction to Mathematical Analysis
Academic year:

2024
Views:

118

Pages:

3
Author:

Kyuwon J.

Key Concepts and Notes Definition of a Cauchy Sequence: A sequence {an } is called a Cauchy sequence if, for every ε > 0, there exists an N ∈ N such that for all n, m ≥ N , we have: |an − am | < ε. This definition means that as the sequence progresses (i.e., for sufficiently large n and m), the terms of the sequence get arbitrarily close to each other, regardless of whether the sequence converges to a specific value or not. — Subsequence: A subsequence {akℓ } of a sequence {an } is obtained by selecting certain elements of the sequence while preserving their original order. Specifically, {kℓ } is a strictly increasing sequence of indices such that k1 < k2 < k3 < . . . , and {akℓ } = {ak1 , ak2 , ak3 , . . . }. For example: • If {an } = {1, 1/2, 1/3, 1/4, . . . }, a possible subsequence is {akℓ } = {a2 , a4 , a6 , . . . } = {1/2, 1/4, 1/6, . . . }. • Another subsequence is {akℓ } = {a1 , a3 , a5 , . . . } = {1, 1/3, 1/5, . . . }. — Key Idea: The goal is to show that if {an } is a Cauchy sequence, then any subsequence {akℓ } of {an } must also be a Cauchy sequence. This holds because: • A subsequence inherits the behavior of the original sequence. • The Cauchy property depends on how close terms are to one another as n, m → ∞. Since the subsequence consists of terms from the original sequence, their closeness is preserved. — Why This is True: 1. **Order of Indices is Preserved:** A subsequence {akℓ } of {an } retains the order of terms in {an }. This means that for sufficiently large ℓ and p, the indices kℓ and kp will also be sufficiently large. 2. **Closeness Condition Holds:** Since {an } is Cauchy, for any ε > 0, there exists N ∈ N such that |an − am | < ε for all n, m ≥ N . For a subsequence {akℓ }, the indices kℓ and kp are also greater than N for sufficiently large ℓ, p. Thus, |akℓ − akp | < ε. — 1 Intuition: Think of a Cauchy sequence as a series of points that get closer together as the sequence progresses. A subsequence picks some of these points while skipping others, but because the original sequence ensures all points are close to each other beyond a certain index, the subsequence also maintains this closeness. — Significance: This property of Cauchy sequences is particularly important because it shows that the Cauchy property is ”inherited” by subsequences. This is foundational in analysis, where Cauchy sequences are used to study convergence in metric spaces and the completeness of the real numbers. — Proof: Every Subsequence of a Cauchy Sequence is also a Cauchy Sequence Let {an } be a Cauchy sequence. By definition, for every ε > 0, there exists an N ∈ N such that for all n, m ≥ N , we have: |an − am | < ε. Let {akℓ } be a subsequence of {an }, where kℓ < kℓ+1 for all ℓ ∈ N. This implies that for all ℓ ∈ N, we have: ℓ ≤ kℓ . For example: • k1 ≥ 1 • k2 > k1 , so k2 ≥ 2 • k3 > k2 , so k3 ≥ 3 Inductively, if p ≤ kp , then p + 1 ≤ kp+1 holds as well. Thus, ℓ ≤ kℓ for all ℓ ∈ N. Since {an } is a Cauchy sequence, for every ε > 0, there exists an N ∈ N such that for all n, m ≥ N , we have: |an − am | < ε. 2 For the same N , we know that for all ℓ, p ≥ N , kℓ ≥ ℓ ≥ N and kp ≥ p ≥ N . Therefore: |akℓ − akp | < ε. This proves that {akℓ } is also a Cauchy sequence. 3