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Thread: Sequence Convergence proof maybe?

  1. #1
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    Sequence Convergence proof maybe?

    a) Suppose that $\displaystyle {X_n}$ and $\displaystyle {Y_n}$ converge to the same point. Prove that $\displaystyle {X_n} - {Y_n} \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.

    b) Show the converse is false.


    Again as usual... some kind of explanation to help me understand this helps alot...
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  2. #2
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    For part a) i would use the $\displaystyle \epsilon-\delta$ definition of limits and it should follow directly. For part b) you just need to show a counter example. Maybe look at $\displaystyle y=x\mbox{ and }y^2-x^2=1.$
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    ok that was how I started it... thanks for the info...
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    Quote Originally Posted by Caity View Post
    a) Suppose that $\displaystyle {X_n}$ and $\displaystyle {Y_n}$ converge to the same point. Prove that $\displaystyle {X_n} - {Y_n} \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.
    b) Show the converse is false.
    Suppose that $\displaystyle L$ is the common limit of the two sequences.
    If $\displaystyle \varepsilon > 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {x_n - L} \right| < \frac{\varepsilon }
    {2}\,\& \,\left| {y_n - L} \right| < \frac{\varepsilon }
    {2}} \right]$.
    Then observe that $\displaystyle \left| {x_n - y_n } \right| \leqslant \left| {x_n - L} \right| + \left| {L - y_n } \right|$.
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    I finished part a). Thanks for the help. Now about part b). First what exactly would the converse be? I know its If X then Y becomes If Y then X. I was thinking maybe by showing something like subtraction is not commutative. Like if X - Y $\displaystyle \rightarrow$ 0 is not the same as Y - X $\displaystyle \rightarrow $0.
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    The converse is: Suppose that $\displaystyle \left( {x_n - y_n } \right) \to 0$ the $\displaystyle \left( {x_n } \right)$ & $\displaystyle \left( {y_n } \right)$ have the same limit.
    Last edited by Plato; Oct 27th 2008 at 01:25 PM.
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    Quote Originally Posted by Plato View Post
    The converse is: Suppose that $\displaystyle \left( {x_n - y_n } \right) \to 0$ the $\displaystyle \left( {x_n } \right)$ & $\displaystyle \left( {y_n } \right)$ have the same limit.

    Well I tried this Plato... but it comes out true and I'm supposed to be proving it false... If the limit of $\displaystyle x_n$ and $\displaystyle y_n$ go to the same number say 1. Then we have 1 - 1 = 0 which shows it is true.
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  8. #8
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    Try this counterexample.
    $\displaystyle x_n = \left( { - 1} \right)^n + \frac{1}
    {n}\,\& \,y_n = \left( { - 1} \right)^n - \frac{1}
    {n}\,$

    Note: we are not given that the sequences converge, only that their difference is null.
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  9. #9
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    Quote Originally Posted by Plato View Post
    Note: we are not given that the sequences converge, only that their difference is null.
    Is it possible to find a counterexample with the two converging ?
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  10. #10
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    Quote Originally Posted by Moo View Post
    Is it possible to find a counterexample with the two converging ?
    The answer is no, the limit must be the same for both sequences.
    Suppose that $\displaystyle \left( {x_n } \right) \to L,\quad \left( {y_n } \right) \to K\,\& \,\left| {\left( {x_n } \right) - \left( {y_n } \right)} \right| \to 0$.
    That means that almost all the terms of $\displaystyle \left( {x_n } \right)$ are ‘close’ to $\displaystyle L$; almost all the terms of $\displaystyle \left( {y_n } \right)$ are ‘close’ to $\displaystyle K$; and almost all the terms of $\displaystyle \left( {x_n } \right) - \left( {y_n } \right)$ are ‘close’ to $\displaystyle 0$.

    Suppose $\displaystyle L \not= K$ then $\displaystyle \varepsilon = \frac{{\left| {L - K} \right|}}
    {4} > 0$.
    But how can almost all of the three sequences be with in an $\displaystyle \epsilon$-distance of its limit?
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