1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Analysis + Cauchy's criterion

  1. Feb 4, 2006 #1
    there is N so that
    [tex] |S_n(x) - S_m(x)| \leq \epsilon [/tex] for ever x in I if n,m N

    ( prove by cauchy's criterion )

    claim: [tex] lim S_n(x) = S(x) [/tex]

    [tex] |S_n(x) - S(x)| < \epsilon /2[/tex] if n[tex] \geq N [/tex]

    then,

    [tex] |S_n(x) - S_m(x)| < |S_n(x) - S(x)| + |S(x) - S_m(x)| [/tex]
    < [tex] \epsilon /2 + \epsilon /2 [/tex]
    < [tex] \epsilon [/tex]

    therefor the series converges pointwise to a funtion S(x)
    im stuck here, I dont know where to go, to say that [tex] S_n(x) [/tex] converges uniformly on I.
     
  2. jcsd
  3. Feb 4, 2006 #2

    JasonRox

    User Avatar
    Homework Helper
    Gold Member

    Have you tried using Triangle Inequalities?
     
  4. Feb 4, 2006 #3
    no, im not sure how that will show uniform convergence??
     
  5. Feb 4, 2006 #4

    JasonRox

    User Avatar
    Homework Helper
    Gold Member

    You are working with inequalities. It is your job it is to show that there is uniform convergence. The Triangle Inequality is a really good tool, especially with limits.

    It's your job to deduce this fact.

    First are you proving Cauchy's Criterion implies that the sequence is convergent?
     
  6. Feb 4, 2006 #5
    yes i used Cauchy's Criterion to show that it converges pointwise
     
  7. Feb 4, 2006 #6

    JasonRox

    User Avatar
    Homework Helper
    Gold Member

    Can you assume that a Cauchy sequence is bounded?

    If not, try proving that first. It would be a great tool to use.
     
  8. Feb 4, 2006 #7
    bounded eh, well ill try that.. ...
     
  9. Feb 5, 2006 #8

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    What is I, what are the S_n? One presumes I is a compact interval, probably [0,1], and that S_n are continuous functions.

    What exactly are you trying to prove?

    As far as I can tell what you wrote states that the S_n converge uniformly.

    I find it impossible to deduce what you've been given and what you're asked to prove.
     
  10. Feb 5, 2006 #9

    JasonRox

    User Avatar
    Homework Helper
    Gold Member

    He's trying to prove that if a sequence is a Cauchy Sequence then the sequence converges to some limit L.

    It's definitely possible, but like you said, probably not with the stuff he's been given.
     
  11. Feb 5, 2006 #10
    yes, thanks jasonrox...thats exactly what im trying to do..but unfortunetly I cant seem to do.
     
  12. Feb 5, 2006 #11

    JasonRox

    User Avatar
    Homework Helper
    Gold Member

    If you haven't proved that it is bounded yet, ignore that. Just move on with the assumption that it is bounded, prove that it is later.

    So, what do you know about bounded sequences?
     
  13. Feb 5, 2006 #12
    well i know that every bounded sequence in the Reals has a convergent subsequence....Also every Cauchy sequence In the Reals conveges.
    and some stuff about Reimman measurable/measure ... which wont help this problem
     
  14. Feb 5, 2006 #13

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    a sequence of what? Functions, we are to assume, I imagine, and continuous ones, probably. It would be nice for that to be stated.

    that is impossible to do since it has not been stated in what space are looking at this Cauchy sequence.
     
  15. Feb 5, 2006 #14

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Let me state what I think the question appears to be:

    let S_n be a cauchy sequence in the sup norm on C([0,1]), prove that S_n converges to a continuous function.
     
  16. Feb 5, 2006 #15
    yes, thats pretty much what im trying to prove.
     
  17. Feb 5, 2006 #16

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Good, but what I wrote bears only passing relation to what you actually stated.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Analysis + Cauchy's criterion
  1. Cauchys criterion (Replies: 4)

Loading...