Exercise · ZW-CONVERGENCE-IN-A-METRIC-SPACE-009 · ★★★ Advanced
Which statement is equivalent to \(x_n\to x\)?
Choose one option.
Show answer
- B. Every ball \(B(x,\varepsilon)\) contains all but finitely many of the \(x_n\)
Explanation
Being in \(B(x,\varepsilon)\) from some \(N\) on is the definition. «Infinitely many» is weaker (it holds for \((-1)^n\) and \(x=1\)), and \(d(x_n,x_{n+1})\to0\) does not guarantee a limit.
Review the concept: Convergence in a metric space
A sequence \((x_n)\) in a metric space \((X,d)\) converges to \(x\in X\) if, for every \(\varepsilon>0\), there is an index \(N\) from which \(x_n\) lies within distance \(\varepsilon\) of \(x\).
\[\forall\varepsilon>0\ \ \exists N:\ n\ge N\Rightarrow d(x_n,x)<\varepsilon\]
More practice problems: convergence in a metric space
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-001 In usual ℝ, what is the limit of x_n=1/n?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-002 For x_n=1/n and ε=0.01 in ℝ, what is the smallest N with d(x_n,0)<ε for all n≥ N?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-003 In Euclidean ℝ^2, to which point does x_n=(tfrac1n,tfracn+1n) converge?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-004 In ℝ with the discrete metric, which of these sequences converges?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-005 In usual ℝ, what happens with x_n=(-1)^n?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-006 If x_n→ x and x_n→ y in a metric space, which argument proves x=y?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-007 Let X=(0,1) with the usual distance and x_n=tfrac1n+1. Does it converge in X?
- ZW-CONVERGENCE-IN-A-METRIC-SPACE-008 In ℝ^2 with d_∞(p,q)=max(|p_1-q_1|,|p_2-q_2|), let x_n=(tfrac1n,tfrac1n^2)→(0,0). What is the small…
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