Exercise · ZW-CAUCHY-SEQUENCE-009 · ★★★ Advanced
A Cauchy sequence \((a_n)\) has a subsequence converging to \(L\). What follows?
Choose one option.
Show answer
- B. \(a_n\to L\) (the whole sequence)
Explanation
Given \(\varepsilon\), terms with large indices are within \(\varepsilon/2\) of each other, and some term of the subsequence is within \(\varepsilon/2\) of \(L\); by the triangle inequality all are within \(\varepsilon\) of \(L\).
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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