Usbrth

Let $X=(mathbbR^k,d)$ denote the standard n-Euclidean space.

Now I read a theorem which states that A sequence $x_n=(x_1,n,x_2,n,x_3,n...x_k,n)$ in $X$ is Cauchy if and only if $x_j,n$ is Cauchy in $R$

But can we generalize this result?

Question : Does this result hold for general metric spaces, and if not, what conditions will guarantee that it does?

I mean take $X_1, d_1$ and $X_2,d_2$ be any two metric space. Let $X=X_1times X_2$ and let $d$ be any metric on $X$( not necessarily a product metric )
then can we say that a $(x_n,x_m)$ is Cauchy in $X=X_1times X_2$ if and only if $x_n$ is Cauchy in $X_1$ and $x_m$ is Cauchy in $X_2$?

As far as I know, being Cauchy is a property of metric, so the result may not hold in general.

Can some one please clarify? Are there some counterexamples.

Thanks a lot.

If you let $d$ be any metric on $X$ then of course the answer is no. Let's say we take $mathbbR$ with the usual metric and $mathbbR^2$ with the discrete metric. (i.e $d(x,y)=delta_xy)$.Then the sequence $(frac1n,frac1n)$ is not Cauchy in $mathbbR^2$ but in both coordinates you have Cauchy sequences.

Taking the discrete metric on $X_1times X_2$ will probably be enough to find a counterexample...

When considering a product of two metric spaces, say $(X_1,d_1)$ and $(X_2,d_2)$, you want to consider metrics on the set $X_1times X_2$ that are naturally related to the individual metrics $d_1,d_2$. For example, you may consider
$d((x,y),(xi,eta)):=d_1(x,xi)+d_2(y,eta)$. More generally, if you have a countable collection of metric spaces $(X_i,d_i)_i=1^infty$ you may define a metric on the product $prod_i=1^inftyX_i$ by
$$d((x_i),(y_i))=sum_i=1^inftyfracd_i(x_i,y_i)2^i$$
For such metrics on the product, which use the individual metrics in some "natural" way, it is the case that $(x_i,n)_n=1^infty$ is a Cauchy sequence in the product space if and only if each sequence $x_i,n$ is a Cauchy sequence in $X_i$.