Results 1 to 4 of 4

Math Help - How to prove that all cauchy sequences are bounded?

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    37

    How to prove that all cauchy sequences are bounded?

    Hi I was just wondering if anyone could help me out a little with this question?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    13
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,984
    Thanks
    1647
    Since \{a_n\} is Cauchy, there exist N such that if m,n are both larger than N, then |a_n- a_m|< 1. Let M be the largest of |a_n- a_m| for m and n both less than or equal to N+1.

    Now, show that |a_n- a_m|< M+1 for all n and m.
    (Consider the cases: (a) n and m both less than N, (b) n and m both larger than N, (c) one of m or n less than N and the other larger.)

    Finally show that for all n, |a_n|< |a_1|+ M+1.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,793
    Thanks
    1688
    Awards
    1
    Quote Originally Posted by binkypoo View Post
    Hi I was just wondering if anyone could help me out a little with this question?
    Here is a neat trick using what Halls has done.
    n \geqslant N\, \Rightarrow \,\left| {x_n  - x_N } \right| < 1\, \Rightarrow \,\left| {x_n } \right| < 1 + \left| {x_N } \right|

    So \left( {\forall k} \right)\left[ {\left| {x_k } \right| < 1 + \sum\limits_{j = 1}^N {\left| {x_j } \right|} } \right]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. All Cauchy Sequences are bounded in a metric space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 1st 2011, 03:18 PM
  2. Cauchy Sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 30th 2009, 04:54 PM
  3. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 21st 2009, 08:59 PM
  4. Replies: 0
    Last Post: July 1st 2008, 02:47 AM
  5. Cauchy/Monotone--Bounded
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 3rd 2007, 06:21 PM

Search Tags


/mathhelpforum @mathhelpforum