Results 1 to 4 of 4

Thread: Bounded and Convergence Proof Question

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    152

    Bounded and Convergence Proof Question

    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.

    Prove the Claim.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,737
    Thanks
    2812
    Awards
    1
    Quote Originally Posted by Janu42 View Post
    assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.
    Claim: If $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.
    Prove the lemma: that there cannot exist two disjoint closed intervals each containing infinitely many points of $\displaystyle a_n$. (Use the given and Bolzano-Weierstrauss)

    By Bolzano-Weierstrauss the set $\displaystyle \left\{a_n:n\in \mathbb{N}\right\}$ has a limit point $\displaystyle L$.

    Use the lemma to prove: if $\displaystyle \varepsilon > 0$ then the interval $\displaystyle \left[ {L - \varepsilon ,L + \varepsilon } \right]$ contains almost all the terms of $\displaystyle a_n$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2010
    Posts
    147
    Quote Originally Posted by Janu42 View Post
    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.

    Prove the Claim.
    I'm probably making this more complicated than it has to be

    Proof by contradiction

    Suppose $\displaystyle (a_{n})$ does not converge to a. Then, there exists an e>0 and a sequence of points $\displaystyle (b_{n})$ such that
    l$\displaystyle b_{n}$ - al > e for all n=1,2,3....

    Look at the sequence $\displaystyle (b_{n})$. By Bolzano-Weierstrauss, it has a convergent subsequence $\displaystyle (c_{n})$, which is also a subsequence of $\displaystyle (a_{n})$. So $\displaystyle (c_{n})$ must converge to a. But l$\displaystyle c_{n}$ - al > e for all n...contradiction.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by Janu42 View Post
    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.

    Prove the Claim.
    I had posted a similar question, and got this response.

    check this thread out:

    http://www.mathhelpforum.com/math-he...bsequence.html
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proof that {(1+x/n)^n} is bounded
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Jan 8th 2012, 02:42 PM
  2. Convergence- proving bounded
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 8th 2010, 03:49 PM
  3. proof that a empty set is bounded
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Sep 17th 2009, 08:31 AM
  4. bounded sequence proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Mar 16th 2007, 09:44 AM
  5. Another bounded proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 11th 2007, 01:33 PM

Search Tags


/mathhelpforum @mathhelpforum