Exercise · ZW-CONVERGENCE-IN-A-METRIC-SPACE-001 · ★☆☆ Basic
In usual \(\mathbb{R}\), what is the limit of \(x_n=1/n\)?
Choose one option.
Show answer
- C. \(0\)
Explanation
\(d(x_n,0)=1/n\) becomes smaller than any \(\varepsilon>0\) when \(n>1/\varepsilon\). The symbol \(\infty\) is not a point of \(\mathbb{R}\).
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-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?
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- ZW-CONVERGENCE-IN-A-METRIC-SPACE-005 In usual ℝ, what happens with x_n=(-1)^n?
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