1. ## Cauchy Sequences

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

2. Consider $\displaystyle (0,\infty)$ with the hereditary topology from $\displaystyle \mathbb{R}$ and check $\displaystyle (1/n)$.

3. 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.

4. ## Cauchy Sequences

Originally Posted by HallsofIvy
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.
I'm new to the forum. I like this example, because it seems like something I may be able to use on a future test. I'm a grad student taking Real Analysis I and need a few tricks in my back pockets. Thanks!
Terry