Exercise · ZW-COMPLETE-METRIC-SPACE-008 · ★★☆ Intermediate
Which of these sequences is Cauchy in \(\mathbb{R}\)?
Choose one option.
Show answer
- D. \(s_n=\sum_{k=1}^{n}2^{-k}\)
Explanation
\(s_n=1-2^{-n}\) converges to \(1\), hence it is Cauchy. \((-1)^n\) oscillates with distance \(2\) between neighbors, and \(n\) and \(H_n\) grow without bound.
Review the concept: Complete metric space
A metric space \(X\) is complete if every Cauchy sequence \((x_n)\) in \(X\) converges to a point \(x\) of \(X\). A sequence is Cauchy if its terms eventually get arbitrarily close to one another.
\[(x_n)\ \text{Cauchy}\ \Longrightarrow\ \exists\,x\in X:\ x_n\to x\]
More practice problems: complete metric space
- ZW-COMPLETE-METRIC-SPACE-001 Which of the following metric spaces, with the usual distance, is complete?
- ZW-COMPLETE-METRIC-SPACE-002 The sequence x_n=1/n is Cauchy in (0,1]. What follows about this space?
- ZW-COMPLETE-METRIC-SPACE-003 The terms 1, 1.4, 1.41, 1.414, … (decimal approximations of √2) are rational and form a Cauchy sequ…
- ZW-COMPLETE-METRIC-SPACE-004 A set X carries the discrete distance: d(x,y)=1 if x≠ y and d(x,x)=0. Is (X,d) complete?
- ZW-COMPLETE-METRIC-SPACE-005 If x_n→ x in a metric space (X,d), which statement is always true?
- ZW-COMPLETE-METRIC-SPACE-006 For x_n=1/n in ℝ, what is the least N such that |x_m-x_n|<0.01 for all m,n≥ N?
- ZW-COMPLETE-METRIC-SPACE-007 Let (x_n) be Cauchy in X with a subsequence converging to x∈ X. What can be guaranteed?
- ZW-COMPLETE-METRIC-SPACE-009 On ℝ define d(x,y)=|arctan x-arctan y| (a genuine distance). Is (ℝ,d) complete?
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