1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequential Compactness

  1. Nov 22, 2011 #1
    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].
     
    Last edited: Nov 22, 2011
  2. jcsd
  3. Nov 22, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Looks ok!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Sequential Compactness
  1. Sequential Compactness (Replies: 8)

  2. Sequential compactness (Replies: 4)

Loading...