Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Complete by taking an arbitrary cauchy sequence

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data
    (1) Prove the space [tex]\ell_\infty[/tex] is complete
    (2)In [tex]\ell_\infty[/tex](R) , let Y be the subspace of all sequences with only finitely many non-0 terms. Prove that Y is not complete.

    3. The attempt at a solution

    (1)I can show that [tex]\ell\infty[/tex] is complete by taking an arbitrary cauchy sequence and showing that xn[tex]\rightarrow[/tex] x

    (2)Im not sure how to go about this.I figure the best wat to show that its not complete is to try to prove that it is complete?but im not sure where to start?
     
  2. jcsd
  3. Feb 23, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Completeness

    The best way to show it's not complete is to find a Cauchy sequence in Y whose limit is not in Y.
     
  4. Feb 24, 2010 #3
    Re: Completeness

    i have to find a sequence in R that doesnt converge in Y? im not sure now to go about this?
     
  5. Feb 24, 2010 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Completeness

    A sequence in Y whose limit is not in Y. I.e. a sequence of sequences with finitely many nonzero terms converging to a sequence that doesn't have that property. Think about it.
     
  6. Feb 24, 2010 #5
    Re: Completeness

    so i want a sequence with finitely many zeros but i want it to converge to 0 right?
    0, 1, 1/2, 1/3, 1/4.... converges to 0 which isnt in Y thus proving that Y isn't complete?
    Is that really all i have to say?
     
  7. Feb 24, 2010 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Completeness

    One sequence is a POINT in Y. To talk about convergence in Y you need a sequence of points in Y. I.e. a sequence of sequences that converges to another sequence.
     
  8. Feb 24, 2010 #7
    Re: Completeness

    yes,sorry

    (0,0,0,....)
    (0,1,0,0..)
    (0,1,1/2,0,0..)
    (0,1,1/2,1/3,0,0..)
    (0,1,1/2,1/3,1/4,0,...)

    is this what you mean by a sequence of sequences?
    Thanks again for all the help
     
  9. Feb 24, 2010 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Completeness

    That's it.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook