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Thread: convergent sequence

  1. #1
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    convergent sequence

    $\displaystyle prove \ that \ if \ a_n \ is \ an \ increasing \ sequence \ , \ and \ b_n \ is \ a \ decreasing$ $\displaystyle \ sequence \ with \ a_n \ \leq \ b_n \ , \ \forall n\in N \ then \ both$
    $\displaystyle a_n \ and \ b_n \ are \ convergent \ sequences$
    $\displaystyle \ \ with \ lim \ a_n \leq lim \ b_n$
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  2. #2
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    Do you know that any bounded monotone sequence converges?
    Do you understand how $\displaystyle b_1$ is an upper bound for the $\displaystyle (a_n)$ sequence?
    What is a lower bound for the $\displaystyle (b_n)$ sequence?
    Now put this together.
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  3. #3
    Member Abu-Khalil's Avatar
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    Suppose $\displaystyle a_n\to a,b_n\to b$ and $\displaystyle a>b$. Hence, $\displaystyle \exists N_1:a-a_n>a-\frac{a+b}{2},\forall n>N_1$ and $\displaystyle \exists N_2:b_n-b>\frac{a+b}{2}-b,\forall n>N_2$. So for any $\displaystyle n\geq \max\left\{N_1,N_2\right\}$ you have $\displaystyle a_n>b_n$.
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