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  1. #1
    is up to his old tricks again! Jhevon's Avatar
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    Elementary Analysis

    I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

    (a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

    Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.

    (b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?
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  2. #2
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    Quote Originally Posted by Jhevon View Post
    I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

    (a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

    Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.
    For m>n consider:

    |Sm - Sn| = |(Sm - Sm-1) + (Sm-1 - Sm-2) + ... + (Sn+1 - Sn)|

    ................<= |Sm - Sm-1| + |Sm-1 - Sm-2| + ... + |Sn+1 - Sn|

    ................< 2^-(m-1) + 2^-(m-2) + ... + 2^-n

    .................= 2^-n { 1 + 2^-1 + 2^-2 + ... 2^-(m-n) }

    .................= 2^-n (1 - 2^[-(m-n)+1]) / (1-2^-1)

    ..................< 2^(-n+1)

    hence as m, n go to infinity |Sm-Sn| -> 0, that is (Sn) is a Cauchy
    sequence and hence converges.


    (b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?
    No, consider the partial sums of the harmonic series.

    Sn = sum 1/r , r=1 .. n

    We know that the harmonic series diverges hence (Sn) diverges, but:

    |Sn+1 - Sn|= 1/(n+1) < 1/n.

    Which provides the required counter example.

    RonL
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    Quote Originally Posted by Jhevon View Post
    I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

    (a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

    Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.

    (b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?
    Are you using, Kenneth Ross book?
    Chapter 11 on Cauchy Sequences and Monotone Sequences.?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Yup, that's the one.

    I think it was chapter 10 though
    Last edited by ThePerfectHacker; June 3rd 2007 at 08:59 PM.
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    Quote Originally Posted by Jhevon View Post
    I think it was chapter 10 though
    Are you by an chance in CCNY?
    Because in the profile it says you are from NY.
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Yes, I am from CCNY. I'm guessing you are as well, what math class are you in?
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    Quote Originally Posted by Jhevon View Post
    Yes, I am from CCNY. I'm guessing you are as well, what math class are you in?
    That is so cool. We are in the same class. MATH 323.
    I guess I see you tomorrow.
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Cool. How's the rest of the hw going?
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    Quote Originally Posted by Jhevon View Post
    Cool. How's the rest of the hw going?
    I finished it. I am a good boy, I do my homework as fast as I can instead of waiting to the last day.
    (It is called Obsessive-Compulsive Disorder).
    Last edited by ThePerfectHacker; February 20th 2007 at 12:44 PM.
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  10. #10
    is up to his old tricks again! Jhevon's Avatar
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    Nope, it's called being smart. Not only are you smart enough to finish the hw, but your free from the stress i'm going through. i'm worried to death about our test next week
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