question on metric spaces

• Apr 8th 2008, 05:37 PM
squarerootof2
question on metric spaces
so this question asks to find all metric spaces in which every sequence is cauchy. hint given is to consider the metric space (S,d) with two distinct pts x and y and to try to construct a non-cauchy sequence. i don't think i'm understanding the application of cauchy sequences in metric spaces. can someone please help me? thanks!
• Apr 8th 2008, 07:23 PM
ThePerfectHacker
Quote:

Originally Posted by squarerootof2
so this question asks to find all metric spaces in which every sequence is cauchy. hint given is to consider the metric space (S,d) with two distinct pts x and y and to try to construct a non-cauchy sequence. i don't think i'm understanding the application of cauchy sequences in metric spaces. can someone please help me? thanks!

If $\displaystyle S$ has two distinct points construct sequence $\displaystyle \left< a,b,a,b,a,b,... \right>$ this cannot be Cauchy because $\displaystyle d(a,b)$ is not arbitrary small.
• Apr 8th 2008, 07:42 PM
squarerootof2
if you don't mind me asking another question, how would the construction of non-cauchy sequence help us find "all" the metric spaces where every sequence is cauchy? would i approach the question with something like, all spaces where the sequences X_n converges (ie metrice spaces where each sequences in the space has a limit)?
• Apr 11th 2008, 01:26 PM
ThePerfectHacker
Quote:

Originally Posted by squarerootof2
if you don't mind me asking another question, how would the construction of non-cauchy sequence help us find "all" the metric spaces where every sequence is cauchy? would i approach the question with something like, all spaces where the sequences X_n converges (ie metrice spaces where each sequences in the space has a limit)?

Because if a metric space has more than 1 point then not all sequences are Cauchy. Thus, only the metric spaces with exactly 1 point have their sequences Cauchy.