Hi,
i tried searching around for informaton on convergent and non-convergent cauchy sequences, but with no success.
anyone to exaplain what is a non-convergent cauchy sequence in a vector space is? examples would also do best.
Hi,
i tried searching around for informaton on convergent and non-convergent cauchy sequences, but with no success.
anyone to exaplain what is a non-convergent cauchy sequence in a vector space is? examples would also do best.
Actually all Cauchy-sequences do converge to something. But that something doesn't have to be in that particular vector space
Example, $\displaystyle \mathbb{Q}$ is a vector space. We can make a cauchy-sequence $\displaystyle (a_n)$ of rational numbers converging to $\displaystyle \sqrt{2}$. The limit is not in $\displaystyle \mathbb{Q}$, thus we say $\displaystyle (a_n)$ does not converge in $\displaystyle \mathbb{Q}$ but it does so in $\displaystyle \mathbb{R}$
Dinkydoe is right. In order to have a Cauchy sequence that does not converge, you must have an incomplete vector space (actually, we're really dealing with metric spaces here, otherwise the notion of distance might not have much meaning!). Dinkydoe's example is of a space that is not complete. If the space is complete, then there's a theorem that says that Cauchy sequences converge, which is an incredibly handy theorem in both pure and applied mathematics.