Results 1 to 6 of 6

Math Help - question about sequences

  1. #1
    Junior Member
    Joined
    Jan 2009
    From
    vancouver
    Posts
    58

    question about sequences

    it's a fact that every convergent sequence is bounded. recall that, by definition, a sequence {An}has the limit L if for every W>0 there is a corresponding integer N such that
    absolute value of (An-L)<W whereever n>N

    Use this definition to prove that every sequence that converges to 0 is bounded, i.e to prove that if lim(n is infinity) An =0 then there is a positive number M such that absolute value of (An)<= M for all n belong to natural numbers.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,957
    Thanks
    1780
    Awards
    1
    Quote Originally Posted by shannon1111 View Post
    it's a fact that every convergent sequence is bounded. recall that, by definition, a sequence {An}has the limit L if for every W>0 there is a corresponding integer N such that
    absolute value of (An-L)<W whereever n>N

    Use this definition to prove that every sequence that converges to 0 is bounded, i.e to prove that if lim(n is infinity) An =0 then there is a positive number M such that absolute value of (An)<= M for all n belong to natural numbers.
    Why are you asked to prove something that you already know is true?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    May 2009
    Posts
    471
    Why not prove the fact first, and then by an obvious corollary, your statement will be true?

    Proposition 3.1.4: Convergent Sequences are Bounded
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jan 2009
    From
    vancouver
    Posts
    58
    Quote Originally Posted by Plato View Post
    Why are you asked to prove something that you already know is true?
    I don't know , my prof ask to do that , that is the assignment for this week
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    Let \{x_n\} be a sequence which converges to 0. Given any epsilon ball about 0, say B(0,\epsilon) you know that there exists N\in \mathbb{Z} such that x_n\in B(0,\epsilon) for all n > N. So simply take

    M=max\{x_1,x_2,...,x_N, \epsilon \} and you can be sure that x_n < M for all n, i.e. it is bounded.
    Last edited by Gamma; July 14th 2009 at 11:16 PM. Reason: fixed disagreement between deltas and epsilons
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Mar 2009
    Posts
    38
    Shannon: While well intentioned, Gamma did you no favor by posting a solution to your problem (no offense intended Gamma but I hope you will think about this too; more hints, less solution will teach more). This most fundamental question and the technique for proving it touches on about every area of mathematics. We've taken away your chance to discover the solution with the hints that you've been given as you've progressed to this point over the years.

    All is not lost. You should write your own proof for a simple case on the real line in about a month without looking at Gamma's proof to make sure you know it and aren't just repeating Gamma's work. I realize that the idea isn't obvious to you now but believe me, it will pay off in your future.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sequences : Question
    Posted in the Pre-Calculus Forum
    Replies: 9
    Last Post: January 7th 2011, 11:28 AM
  2. Sequences question!
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 17th 2010, 07:02 PM
  3. Am I right? (Sequences question)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 3rd 2010, 03:33 PM
  4. Sequences question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 17th 2009, 12:13 AM
  5. Sequences Question Please Help
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 2nd 2009, 12:33 PM

Search Tags


/mathhelpforum @mathhelpforum