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Thread: null sequences.

  1. #1
    Senior Member abhishekkgp's Avatar
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    null sequences.

    Prove or disprove:
    If is a null sequence then one or both of is(are) null.

    My attempt:
    if both are null then certainly is null, no problem with that.

    I assumed that is not null, is not null and as given, is null. I tried to bring a contradiction since i have a feeling that the statement in the question is true.

    since its assumed that is not null so:
    there exists such that for all there exists such that

    similarly:
    there exists such that for all there exists such that

    and since is null:
    there exists such that

    now since i can't say that occur at the same i can't find a contradiction.

    someone help.
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    Quote Originally Posted by abhishekkgp View Post
    Prove or disprove:
    If is a null sequence then one or both of is(are) null.
    Assuming both converge, suppose that neither is a null sequence.
    Then $\displaystyle (x_n)\to X\ne 0$ and $\displaystyle (y_n)\to Y\ne 0$.
    What can you say about $\displaystyle (x_ny_n)\to~ ?$
    What does that tell you.
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Plato View Post
    Assuming both converge, suppose that neither is a null sequence.
    Then $\displaystyle (x_n)\to X\ne 0$ and $\displaystyle (y_n)\to Y\ne 0$.
    What can you say about $\displaystyle (x_ny_n)\to~ ?$
    What does that tell you.
    probably and probably null sequences converge to 0 and hence a contradiction. But convergence has not yet been discussed in the book i am reading(first course in real analysis- stirling K berberian) so i can only guess.
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    Quote Originally Posted by abhishekkgp View Post
    But convergence has not yet been discussed in the book i am reading(first course in real analysis- stirling K berberian) so i can only guess.
    How very odd. I know Berberian's book on measure theory and find it odd.
    Pray tell then what is a null sequence?
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    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Plato View Post
    How very odd. I know Berberian's book on measure theory and find it odd.
    Pray tell then what is a null sequence?

    a sequence in is said to be null if:
    for all there exists such that
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    Quote Originally Posted by abhishekkgp View Post
    a sequence in is said to be null if:
    for all there exists such that
    OK go back to the OP.
    Let $\displaystyle \varepsilon = \frac{{\varepsilon _1 \varepsilon _2 }}{2}$.
    Make $\displaystyle |x_ny_n|<\varepsilon$ then $\displaystyle \varepsilon _1 \varepsilon _2\le |x_n||y_n|=|x_ny_n|<\varepsilon$.
    That is a contradiction.
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  7. #7
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Plato View Post
    OK go back to the OP.
    Let $\displaystyle \varepsilon = \frac{{\varepsilon _1 \varepsilon _2 }}{2}$.
    Make $\displaystyle |x_ny_n|<\varepsilon$ then $\displaystyle \varepsilon _1 \varepsilon _2\le |x_n||y_n|=|x_ny_n|<\varepsilon$.
    That is a contradiction.
    the problem is that i can't( or yet not able to) say that there exists an such that . I can only say that and that's why i can't contradict anything.
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  8. #8
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    Quote Originally Posted by abhishekkgp View Post
    the problem is that i can't( or yet not able to) say that there exists an such that . I can only say that and that's why i can't contradict anything.
    I think that I misread the OP.
    Consider this example.
    $\displaystyle x_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is even}} \\ {\frac{1}{n},} & {\text{ n is odd}} \\ \end{array} } \right.$ and $\displaystyle y_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is odd}} \\ {\frac{1}{n},} & {\text{ n is even}} \\ \end{array} } \right.$
    What is $\displaystyle x_ny_n~?$
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  9. #9
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Plato View Post
    I think that I misread the OP.
    Consider this example.
    $\displaystyle x_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is even}} \\ {\frac{1}{n},} & {\text{ n is odd}} \\ \end{array} } \right.$ and $\displaystyle y_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is odd}} \\ {\frac{1}{n},} & {\text{ n is even}} \\ \end{array} } \right.$
    What is $\displaystyle x_ny_n~?$
    so it can happen that if we have $\displaystyle (x_n)$ not null, $\displaystyle (y_n)$ not null, we still have $\displaystyle (x_ny_n)$ as null!!

    thank you for this.
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