Prove that if metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete so is metric space $X\times Y$ complete

Question

a) Metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete. Prove that metric space $X\times Y$ is complete if the metric is defined as:

$\rho((x_1,y_1),(x_2,y_2))$ = $\sqrt{d_X(x_1,x_2)^2 + d_Y(y_1,y_2)^2}$

b) Prove that if $(X_1,d_{X1})$, $(X_2,d_{X2})$$, ...$ are complete then product $X_1\times X_2\times ...$ is a complete space when the distance between points $x=(x_1,x_2,...)$, $y=(y_1,y_2,...)$ is defined as:
$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{2^n} \frac{d_n(x_n,y_n)}{1+d(x_n,y_n)} $

Answer 1

Here's the proof outline:

Suppose we have a Cauchy sequence $x^{(1)}, x^{(2)}, \ldots$ in $\prod X_i$.

Then each $x^{(n)}$ can be written as $(x^{(n)}_1, x^{(n)}_2,\ldots)$ for each $x^{(n)}_i \in X_i$.

Exercise: Prove for each fixed $i$ the sequence $x^{(n)}_i$ is Cauchy in $X_i$.

By completeness of $X_i$ we have $x^{(n)}_i \to y_i$ for some $y_i \in X_i$.

Define the element $y = (y_1,y_2,\ldots)$ of $\prod X_i$.

Exercise: Prove $x^{(n)} \to y$ in $\prod X_i$.

We conclude each Cauchy sequence in $\prod X_i$ is convergent.