Completeness Practice Problem and Solution

Practice Problems and Step-by-Step Solutions Problem 1: Prove that R is complete. Step-by-Step Solution: 1. Let {an } be a Cauchy sequence in R. 2. By the definition of a Cauchy sequence, for every ε > 0, there exists N ∈ N such that for all n, m ≥ N : |an − am | < ε. 3. Since {an } is a Cauchy sequence, it is bounded. That is, there exists M > 0 such that |an | ≤ M for all n. 4. By the Bolzano-Weierstrass theorem, every bounded sequence in R has a convergent subsequence. Let {ank } be a subsequence of {an } that converges to some L ∈ R. 5. To show that {an } converges to L, observe that for any ε > 0, there exists N ∈ N such that: |an − ank | < ε/2 and |ank − L| < ε/2. 6. By the triangle inequality: |an − L| ≤ |an − ank | + |ank − L| < ε/2 + ε/2 = ε. 7. Thus, {an } converges to L, and L ∈ R. Therefore, R is complete. — Problem 2: Show that Q is not complete. Step-by-Step Solution: 1. Consider the sequence {an } = 3, 3.1, 3.14, 3.141, . . . in Q, which approximates π. 2. Observe that for any ε > 0, there exists N ∈ N such that for all n, m ≥ N : |an − am | < ε. This shows that {an } is a Cauchy sequence in Q. 3. However, the limit of {an } is π, which is not in Q. 4. Since {an } does not converge to a point in Q, the space Q is not complete. — Problem 3: Prove that the interval [0, 1] is complete. Step-by-Step Solution: 1. Let {an } be a Cauchy sequence in [0, 1]. 2. Since {an } is Cauchy, it is bounded. Thus, 0 ≤ an ≤ 1 for all n. 1 3. By the Bolzano-Weierstrass theorem, every bounded sequence in R has a convergent subsequence. Let {ank } be a subsequence of {an } that converges to some L ∈ R. 4. Since 0 ≤ ank ≤ 1 for all k, the limit L must also satisfy 0 ≤ L ≤ 1. 5. To show that {an } converges to L, for any ε > 0, there exists N ∈ N such that: |an − ank | < ε/2 and |ank − L| < ε/2. 6. By the triangle inequality: |an − L| ≤ |an − ank | + |ank − L| < ε/2 + ε/2 = ε. 7. Thus, {an } converges to L ∈ [0, 1], proving that [0, 1] is complete. — Practice Problems 1. Prove that every closed subset of R is complete. 2. Show that the open interval (0, 1) is not complete. 3. Construct an example of a Cauchy sequence in Q that does not converge. 2