Connection Between Cauchy and Convergent Sequences

Connection Between Cauchy and Convergent Sequences Key Concepts Definition of a Cauchy Sequence: A sequence {an } is called a Cauchy sequence if, for every ε > 0, there exists N ∈ N such that for all n, m ≥ N , we have: |an − am | < ε. This definition means that as the sequence progresses, the terms of the sequence become arbitrarily close to each other, irrespective of whether the sequence converges to a specific value. Definition of a Convergent Sequence: A sequence {an } is called convergent if there exists a limit L ∈ R such that for every ε > 0, there exists N ∈ N such that for all n ≥ N , we have: |an − L| < ε. This means the terms of the sequence approach L as n → ∞. — Relationship Between Cauchy and Convergent Sequences Theorem 1: Every Convergent Sequence is Cauchy. If {an } converges to L, then {an } is a Cauchy sequence. Proof: 1. Since {an } converges to L, for every ε > 0, there exists N ∈ N such that for all n ≥ N , we have: ε |an − L| < . 2 2. For n, m ≥ N , using the triangle inequality: |an − am | ≤ |an − L| + |L − am |. 3. Since |an − L| < ε 2 and |L − am | < 2ε , we have: |an − am | < ε ε + = ε. 2 2 4. Thus, {an } satisfies the definition of a Cauchy sequence. Theorem 2: Every Cauchy Sequence in a Complete Metric Space is Convergent. If {an } is a Cauchy sequence in a complete metric space (e.g., R), then {an } converges to some L ∈ R. — 1 Importance of Completeness The property of completeness ensures that a Cauchy sequence converges to a limit within the space. A metric space is complete if every Cauchy sequence in the space converges to a limit that belongs to the space. For example: • R (the real numbers) is complete. Every Cauchy sequence in R has a limit in R. • Q (the rational numbers) is not complete. There are Cauchy sequences in Q, such as {3, 3.1, 3.14, 3.141, . . . } (approximating π), that do not converge in Q because π ∈ / Q. Key Idea: Completeness is necessary to ensure the existence of a limit. Without completeness, Cauchy sequences may ”escape” the space. — Examples 1. **Example 1: A Convergent Sequence is Cauchy** Consider the sequence {an } = which converges to L = 0. 1 , n • For any ε > 0, choose N > 1ε . For all n, m ≥ N , we have: |an − am | = 1 1 1 ≤ − < ε. n m N • Since {an } converges to 0, it also satisfies the definition of a Cauchy sequence. 2. **Example 2: A Cauchy Sequence in Q That Does Not Converge in Q** Consider the sequence {an } = 3, 3.1, 3.14, 3.141, . . . , which approximates π. • The sequence is Cauchy because for any ε > 0, we can find N such that for all n, m ≥ N , the terms are sufficiently close. • However, π ∈ / Q, so the sequence does not converge in Q. • If we extend the space to R, the sequence converges to π. 3. **Example 3: A Constant Sequence** The sequence {an } = c, where c is a constant, is trivially both convergent and Cauchy because: |an − am | = |c − c| = 0 for all n, m. — 2 Practice Problems Problem 1: Show that the sequence {an } = 1 n is both convergent and Cauchy. Solution: • The sequence {an } converges to L = 0 because for any ε > 0, we can choose N > such that for all n ≥ N , we have: |an − 0| = 1 ε 1 < ε. n • The sequence is Cauchy because for all n, m ≥ N , we have: |an − am | = 1 1 1 − ≤ < ε. n m N Problem 2: Provide an example of a sequence in Q that is Cauchy but does not converge in Q. Solution: The sequence {an } = 3, 3.1, 3.14, 3.141, . . . , approximating π, is Cauchy in Q because: • For any ε > 0, we can choose N such that for all n, m ≥ N , the terms satisfy |an −am | < ε. • However, the sequence does not converge in Q because π ∈ / Q. Problem 3: Show that every constant sequence is Cauchy. Solution: If {an } = c for some constant c, then for all n, m: |an − am | = |c − c| = 0. Hence, the sequence is trivially Cauchy. — 3