Results 1 to 3 of 3

Thread: sequence, convergence

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    17

    sequence, convergence

    Let $\displaystyle (x_n)$ be a sequence of real numbers.

    (a) Prove that $\displaystyle (x_n)$ converges to a number $\displaystyle A$ iff every subsequence of $\displaystyle (x_n)$ has a subsequence which converges to $\displaystyle A$.
    (b) Does $\displaystyle (x_n)$ converge if every subsequence of $\displaystyle (x_n)$ has a subsequence which converges?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Dec 2008
    Posts
    17
    I figured out part b, but part a is confusing me.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by poincare4223 View Post
    (a) Prove that $\displaystyle (x_n)$ converges to a number $\displaystyle A$ iff every subsequence of $\displaystyle (x_n)$ has a subsequence which converges to $\displaystyle A$.
    If $\displaystyle \{x_n\}$ is convergent then all subsequences converge to the same limit. Thus, the forward direction follows.

    To prove the backwards direction assume that $\displaystyle \{x_n\}$ [u]did not[/b] converge to $\displaystyle A$. Then it means there is some $\displaystyle \epsilon > 0$ so that for any $\displaystyle N>0$ we have $\displaystyle |x_n - A| \geq \epsilon$ for some $\displaystyle n>N$. Thus, we can form a subsequence, $\displaystyle \{x_{n_k}\}$ with $\displaystyle |x_{n_k} - A|\geq \epsilon$. But this subsequence cannot possible have a subsequence converging to $\displaystyle A$ . Contradiction.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. convergence of a sequence
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: Mar 24th 2011, 03:11 PM
  2. Replies: 7
    Last Post: Oct 12th 2009, 10:10 AM
  3. Replies: 6
    Last Post: Oct 1st 2009, 09:10 AM
  4. Convergence of sequence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Sep 23rd 2009, 05:33 AM
  5. Replies: 6
    Last Post: Oct 24th 2008, 01:45 PM

Search Tags


/mathhelpforum @mathhelpforum