Exercises
[0VD]Let be given a metric space \((X,d)\). As in [0NW] we define the disk \(D(x,\varepsilon ){\stackrel{.}{=}}\{ y∈ X, d(x,y)≤ \varepsilon \} \) (which is closed). \((X,d)\) is locally compact if for every \(x∈ X\) there exists \(\varepsilon {\gt}0\) such that \(D(x,\varepsilon )\) is compact. Consider this proposition.
«Proposition A locally compact metric space is complete. Proof Let \((x_ n)_ n⊂ X\) be a Cauchy sequence, then eventually its terms are distant at most \(\varepsilon \), so they are contained in a small compact disk, so there is a subsequence that converges, and then, by the result [0N8], the whole sequence converges. q.e.d. »If you think the proposition is true, rewrite the proof rigorously. If you think it’s false, find a counterexample.
Solution 1