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Math Help - Sequences

  1. #1
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    Sequences

    Let (xn) and (yn) be two sequences. Let (zn) be the shuffled sequence:
    z1=x1, z2=y1, z3=x2, z4=y2.....z(2n-1)=xn, z(2n)=yn...

    Prove that (zn) converges, which implies that both (xn) and (yn) converge.

    Can someone help?
    Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JaysFan31 View Post
    Let (xn) and (yn) be two sequences. Let (zn) be the shuffled sequence:
    z1=x1, z2=y1, z3=x2, z4=y2.....z(2n-1)=xn, z(2n)=yn...

    Prove that (zn) converges, which implies that both (xn) and (yn) converge.

    Can someone help?
    Thanks.
    Counterexample:
    Let
    x_n = n
    y_n = n^2

    None of the sequences x, y, z converge.

    Do you mean the question to say that "if {z_n} converges then {x_n}, {y_n} converge"?

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    Do you mean the question to say that "if {z_n} converges then {x_n}, {y_n} converge"?
    Yes.

    ------
    I did not try to solve this problem but I am thinking:
    If the sequence {z_n} converges then the odd and even subsequences converge that is,
    z_1,z_3,z_5,... -----> Converges
    z_2,z_4,z_6,... -----> Converges

    But the odd-even subsequences are the sequences x_n and y_n. Thus, {x_n} and {y_n} both converges.

    Is that not true?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by topsquark View Post

    Do you mean the question to say that "if {z_n} converges then {x_n}, {y_n} converge"?
    If so, use the fact that a sequence converges iff every subsequence
    converges then the result is trivial. So maybe we should conclude that
    you don't know this principal?

    RonL
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