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.