A sequence $(x_n,x_m)$ is Cauchy in $X=X_1\times X_2$ if and only if $x_n$ is Cauchy in $X_1$ and $x_m$ is Cauchy in $X_2$? Let $X=(\mathbb{R}^k,d)$ 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 $X=X_1\times X_2$ 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 $X=X_1\times X_2$ 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. 
 A: If you let $d$ be any metric on $X$ then of course the answer is no. Let's say we take $\mathbb{R}$ with the usual metric and $\mathbb{R^2}$ with the discrete metric. (i.e $d(x,y)=\delta_{xy})$.Then the sequence $(\frac{1}{n},\frac{1}{n})$ is not Cauchy in $\mathbb{R^2}$ but in both coordinates you have Cauchy sequences. 
A: Taking the discrete metric on $X_1\times X_2$ will probably be enough to find a counterexample...
A: 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 $X_1\times X_2$ 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 $\prod_{i=1}^{\infty}X_i$ by
$$d((x_i),(y_i))=\sum_{i=1}^{\infty}\frac{d_i(x_i,y_i)}{2^i}$$
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 $X_i$.
