Exercise · ZW-COMPLETE-METRIC-SPACE-003 · ★☆☆ Basic
The terms 1, 1.4, 1.41, 1.414, … (decimal approximations of \(\sqrt2\)) are rational and form a Cauchy sequence . Which statement is correct?
Choose one option.
A It does not converge in \(\mathbb{Q}\), but it does in \(\mathbb{R}\). B It converges in \(\mathbb{Q}\), but not in \(\mathbb{R}\). C It converges in \(\mathbb{Q}\) to 1.4142. D It converges neither in \(\mathbb{Q}\) nor in \(\mathbb{R}\). Show answer A. It does not converge in \(\mathbb{Q}\), but it does in \(\mathbb{R}\).← Back to the question
Explanation It converges in \(\mathbb{R}\) to \(\sqrt2\), which is irrational; hence \(\mathbb{Q}\) is not complete.
A metric space \(X\) is complete if every Cauchy sequence \((x_n)\) in \(X\) converges to a point \(x\) of \(X\). A sequence is Cauchy if its terms eventually get arbitrarily close to one another.
\[(x_n)\ \text{Cauchy}\ \Longrightarrow\ \exists\,x\in X:\ x_n\to x\]
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