# Cauchy Sequences

• Oct 29th 2009, 01:41 AM
Cauchy Sequences
Could anyone give me a specific example of Cauchy Sequence in Metric Space X that does not converge?
• Oct 29th 2009, 02:02 AM
Rebesques
Consider $\displaystyle (0,\infty)$ with the hereditary topology from $\displaystyle \mathbb{R}$ and check $\displaystyle (1/n)$.
• Oct 29th 2009, 07:44 AM
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.
• Nov 30th 2009, 04:54 PM
talb1mnk7
Cauchy Sequences
Quote:

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(Happy)