Exercise · ZW-COMPLETE-METRIC-SPACE-004 · ★☆☆ Basic
A set \(X\) carries the discrete distance: \(d(x,y)=1\) if \(x\neq y\) and \(d(x,x)=0\). Is \((X,d)\) complete?
Choose one option.
A No: no Cauchy sequence converges. B Only if \(X\) is finite. C Yes: every Cauchy sequence is eventually constant, hence it converges. D No: the distance takes only two values. Show answer C. Yes: every Cauchy sequence is eventually constant, hence it converges.← Back to the question
Explanation With \(\varepsilon=1/2\), \(d(x_m,x_n)<\varepsilon\) forces \(x_m=x_n\); the sequence is eventually constant and converges to that value.
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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