Could anyone give me a specific example of Cauchy Sequence in Metric Space X that does not converge?

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- Oct 29th 2009, 01:41 AMfelixmcgradyCauchy Sequences
Could anyone give me a specific example of Cauchy Sequence in Metric Space X that does not converge?

- Oct 29th 2009, 02:02 AMRebesques
Consider $\displaystyle (0,\infty)$ with the hereditary topology from $\displaystyle \mathbb{R}$ and check $\displaystyle (1/n)$.

- Oct 29th 2009, 07:44 AMHallsofIvy
Or, let X be the set of

**rational**numbers with the "usual metric", d(x,y)= |x-y|. Take the sequence 3, 3.1, 3.14, 3.1415, 3.14159, 3.141592, ... where the nth term is the decimal expansion of $\displaystyle \pi$ to n-1 decimal places. It is easy to see that this sequence is a Cauchy sequence and that it does not converge in the rational numbers. - Nov 30th 2009, 04:54 PMtalb1mnk7Cauchy Sequences