Exercise · ZW-CAUCHY-SEQUENCE-002 · ★☆☆ Basic
For \(a_n=1/n\), what is the smallest \(N\) such that \(|a_m-a_n|<0.01\) whenever \(m,n\ge N\)?
Choose one option.
Show answer
- B. \(N=100\)
Explanation
If \(m,n\ge N\), then \(|1/m-1/n|<1/N\). With \(N=100\) the bound is \(0.01\). With \(N=99\) it fails: \(m=99\) and a large \(n\) give a difference close to \(1/99>0.01\).
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.
\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]
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