(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [itex](X,d_X)[/itex] and [itex](Y,d_Y)[/itex] are sequentially compact metric spaces. Show that [itex](X\times Y, d_{X\times Y})[/itex] is sequentially compact where [tex]d_{X\times Y} ((x_1,y_1),(x_2,y_2)) = d_X(x_1,x_2) + d_Y(y_1,y_2)[/tex] is the product metric.

3. The attempt at a solution

Suppose [itex](x_n)_{n\in\mathbb{N}}[/itex] is a sequence in [itex]X[/itex] and [itex](y_n)_{n\in\mathbb{N}}[/itex] is a sequence in [itex]Y[/itex].

Then since [itex]X,Y[/itex] are sequentially compact [itex](x_n)_{n\in\mathbb{N}}[/itex] and [itex](y_n)_{n\in\mathbb{N}}[/itex] have convergent subsequences, say [itex]x_{n_k} \to x\in X[/itex] and [itex]y_{n_k} \to y\in Y[/itex] as [itex]k\to\infty[/itex].

It follows that if [itex](x_n,y_n)_{n\in\mathbb{N}}[/itex] is a sequence in [itex]X\times Y[/itex] with subsequence [itex](x_{n_k},y_{n_k})_{k\in\mathbb{N}}[/itex] then [itex](x_{n_k},y_{n_k}) \to (x,y)\in X\times Y[/itex] as [itex]k\to\infty[/itex].

Is this OK so far? Do I now need to show that [itex](x_{n_k},y_{n_k}) \to (x,y)[/itex] in the metric [itex]d_{X\times Y}[/itex] ?

In which case:

[itex]d_{X\times Y}((x_{n_k} , y_{n_k}),(x,y)) = d_X(x_{n_k} , x) + d_Y(y_{n_k},y) \to 0+0=0[/itex]

so [itex](x_{n_k},y_{n_k}) \to (x,y)[/itex] in the metric [itex]d_{X\times Y}[/itex].

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Sequential Compactness

**Physics Forums | Science Articles, Homework Help, Discussion**