If $X\times Y$ is complete metric space then $X$ and $Y$ are complete metric spaces

Question

If $(X\times Y, D)$ is a complete metric space where $D=d_{X}+d_{Y}$

How to prove that $X$ and $Y$ are also complete ?

Can i say: let $(x_n)$ a Cauchy sequence in $X$ then it is Cauchy in $X\times Y$, so $(x_n)$ converge in $X\times Y$?

Answer 1

Let $y\in Y$, $(x_n,y)$ is a Cauchy sequence implies that $(x_n)$ converges.

Answer 2

I suppose $X$ and $Y$ are nonempty. Let $y_0\in Y$. If $(x_n)$ is a Cauchy sequence in $X$, then $((x_n,y_0))$ is a Cauchy sequence in $X\times Y$ which must converge in $X\times Y\ldots$

Answer 3

Assuming $X$ and $Y$ not empty, let $y\in Y$. Consider the map $$ \varphi\colon X\to X\times Y,\qquad \varphi(x)=(x,y) $$ Then, for $x_1,x_2\in X$, $$ D(\varphi(x_1),\varphi(x_2))= D((x_1,y),(x_2,y))=d_X(x_1,x_2)+d_Y(y,y)=d_X(x_1,x_2) $$ Therefore $\varphi$ is an isometry of $X$ onto a closed (prove it) subspace of $X\times Y$.