What is the Metric for Convergence in Cartesian Product of Metric Spaces?

[CornMuffin]

Homework Statement

Let (X_k,d_k), 1\leq k<\infty be metric spaces
Let X=\prod _{k=1}^{\infty} X_k be their Cartesian product,
that is, let X be the set of sequences (x_1,x_2,...), where x_j\in X_j for 1\leq j < \infty

Show that a sequence \left\{x^{(k)}\right\}_{k=1}^{ \infty } converges in X if and only if \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } converges in X_j for each j \geq 1.

Homework Equations

The Attempt at a Solution

Assume \left\{x^{(k)}\right\}_{k=1}^{ \infty } converges in X.
then \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),... where a_{ik} \in X_i
So, \left\{x^{(k)}\right\}_{k=1}^{ \infty } converges to c_i

Assume \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } converges in X_j for each j \geq 1
so, \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),... where \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } = a_{j1},a_{j2},... converges to c_j
So, \left\{x^{(k)}\right\}_{k=1}^{ \infty } converges to (c_1,c_2,...)

 

[jbunniii]

Can you tell us what is the metric on X?

 

[CornMuffin]

Can you tell us what is the metric on X?

I don't think it gives an explicit metric on X, but from part a (this is part b), it asks to show that something is a metric on X, but I didn't think that carried over (but maybe it does)

from part a:
Show that

d(x,y) = \sum _{j=1}^{ \infty }\frac{1}{2^j}min(1,d_j(x_j,y_j))

is a metric on X

(idk why it kept isametric in the tex tags :(

 

[jbunniii]

I don't think it gives an explicit metric on X, but from part a (this is part b), it asks to show that something is a metric on X, but I didn't think that carried over (but maybe it does)

from part a:
Show that
d(x,y) = \sum _{j=1}^{ \infty }\frac{1}{2^j}min(1,d_j(x_j,y_j))

is a metric on X

I think it must carry over, because there's more than one way to define a metric on X and it's hard to talk about convergence in a metric space if you don't specify what the metric is. Try proceeding under that assumption and let us know if you get stuck.