Exercise · ZW-CAUCHY-SEQUENCE-008 · ★★☆ Intermediate
Why is \(a_n=(-1)^n\) not Cauchy?
Choose one option.
Show answer
- A. For \(\varepsilon=1\) and any \(N\) there are \(m,n\ge N\) with \(|a_m-a_n|=2\)
Explanation
Two consecutive terms are always at distance \(2\ge1\), so no \(N\) makes all distances less than \(1\). Monotonicity or positivity are irrelevant; and \(|a_{n+1}-a_n|=2\) does not tend to \(0\).
Review the concept: Cauchy sequence
A sequence \((a_n)\) is Cauchy if, for every \(\varepsilon>0\), there is an index \(N\) beyond which any two terms are less than \(\varepsilon\) apart.
\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]
More practice problems: cauchy sequence
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