Complete product of metric spaces Prove that: 
If a product $X\times Y$ of metric spaces $(X,\rho_X)$ and $(Y, \rho_Y)$ with metric $\rho((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ is complete, then metrics $\rho_X$ and $\rho_Y$ are complete. Generalize the result for more factors.
I guess that I need to use the fact, that every Cauchy sequence is convergent. So for all $k,m>n$ $\rho((x_k,y_k),(x_m,y_m))=\sqrt{(x_k-y_k)^2+(x_m-y_m)^2}<\epsilon$. So can I say that $(x_k-y_k)^2<\epsilon/2$ and $(x_m-y_m)^2<\epsilon/2$. How does that give me completeness of $\rho_X$ and $\rho_Y$?
 A: Actually I can think of a rather annoying counterexample: Take $X$ a metric space which is not complete and $Y=\emptyset$. Then $X\times Y=\emptyset$ which is vacuously complete (there are no Cauchy sequences), yet $X$ is not.
But it's try assuming neither $X$ nor $Y$ is empty. I'll just show $X$ is complete.
Let $\{x_n\}$ be a Cauchy sequence in $X$, and fix $y \in Y$. Then it's easy to see that $\{(x_n,y)\}$ is a Cauchy sequence in the product, hence it converges to some $(x_0,y_0)$. But this means that $x_n \to x_0$ in $X$.
A: To prove completeness of $\rho_X$, you should start with an arbitrary Cauchy sequence $\langle x_n:n\in\Bbb N$ in $X$, not with a sequence in $X\times Y$.
HINT: Let $\langle x_n:n\in\Bbb N$ be a Cauchy sequence in $X$, and let $y$ be a point of $Y$. Show that $\big\langle\langle x_n,y\rangle:n\in\Bbb N\big\rangle$ is Cauchy in $X\times Y$ and use the fact that $\rho$ is complete to show that $\langle x_n:n\in\Bbb N$ converges.
The proof for $\rho_Y$ will be entirely similar, and the result and proof will generalize easily to products of any finite number of factors.
