Exercise · ZW-COMPLETE-METRIC-SPACE-005 · ★☆☆ Basic
If \(x_n\to x\) in a metric space \((X,d)\), which statement is always true?
Choose one option.
Show answer
- B. \((x_n)\) is a Cauchy sequence.
Explanation
If \(d(x_n,x)<\varepsilon/2\) for \(n\ge N\), the triangle inequality gives \(d(x_m,x_n)<\varepsilon\). The converse is exactly what completeness demands.
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-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-008 Which of these sequences is Cauchy in ℝ?
- 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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