Results 1 to 2 of 2

Thread: prove a lemma for a metric space

  1. #1
    Junior Member hercules's Avatar
    Joined
    Sep 2007
    From
    Earth
    Posts
    62

    prove a lemma for a metric space

    I need help proving this

    Lemma: A sequence $\displaystyle \left( \bold{ x^{(n)}} \right)$ in $\displaystyle \mathbb{R}^k$ converges if and only if for each $\displaystyle j = 1,2,...,k,$ the sequence $\displaystyle \left( x_j^{(n)}\right)$ converges in $\displaystyle \mathbb{R}$.

    Thanks guys
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by hercules View Post
    I need help proving this

    Lemma: A sequence $\displaystyle \left( \bold{ x^{(n)}} \right)$ in $\displaystyle \mathbb{R}^k$ converges if and only if for each $\displaystyle j = 1,2,...,k,$ the sequence $\displaystyle \left( x_j^{(n)}\right)$ converges in $\displaystyle \mathbb{R}$.

    Thanks guys
    Say $\displaystyle |\bold{x}_k - \bold{y} | <\epsilon$.
    Then it means,
    $\displaystyle \sqrt{(x^{(1)}_k - y_1)^2+...+(x^{(n)} - y_k)^2} < \epsilon$
    But that means,
    $\displaystyle \max | x^{(i)}_k - y_i| < \epsilon$.
    Thus, each component converges.

    Conversely if each component converges it means,
    $\displaystyle \max |x^{(i)}_k - y_i| < \epsilon$.
    That means,
    $\displaystyle \sqrt{(x^{(1)}_k - y_1)^2+...+(x^{(n)} - y_k)^2} \leq \sqrt{ n \max (x^{(i)}_k - y_i)^2} = \sqrt{n}\max | x^{(i)} - y_i| < \sqrt{n} \epsilon$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Jul 8th 2011, 02:16 PM
  2. Replies: 1
    Last Post: Oct 30th 2010, 01:50 PM
  3. Limit of function from one metric space to another metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Sep 17th 2010, 02:04 PM
  4. Sets > Metric Space > Euclidean Space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Apr 25th 2010, 10:17 PM
  5. Prove a metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Sep 27th 2009, 09:53 AM

Search Tags


/mathhelpforum @mathhelpforum