Exercise · ZW-CAUCHY-SEQUENCE-010 · ★★★ Advanced

Let \(a_n\) be \(\sqrt2\) truncated to \(n\) decimals: \(1,\,1.4,\,1.41,\,1.414,\dots\). Viewed in \(\mathbb Q\), what happens?

Choose one option.

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C. It is Cauchy but does not converge in \(\mathbb Q\)

Explanation

For \(m>n\) we have \(0\le a_m-a_n<10^{-n}\), so it is Cauchy. Its natural limit, \(\sqrt2\), is irrational, so there is no limit in \(\mathbb Q\): \(\mathbb Q\) is not complete.

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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.

\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]

Let \(a_n\) be \(\sqrt2\) truncated to \(n\) decimals: \(1,\,1.4,\,1.41,\,1.414,\dots\). Viewed in \(\mathbb Q\), what happens?

Explanation

Review the concept: Cauchy sequence

More practice problems: cauchy sequence