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Thread: Convergence of a sequence

  1. #1
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    Convergence of a sequence

    Hey all,

    Unsure on how to structure the proof to the following:

    Suppose that the sequence (an) tends to the limit A, while the sequence (bn) tends to the limit B. Prove that the sequence (an + bn) tends to A + B.

    Any hints would be greatly appreciated.
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  2. #2
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    [QUOTE=scorpio1;113799]Hey all,

    Unsure on how to structure the proof to the following:

    Suppose that the sequence (an) tends to the limit A, while the sequence (bn) tends to the limit B. Prove that the sequence (an + bn) tends to A + B.

    Pick a natural number $\displaystyle N_1$ so for all n > $\displaystyle N_1$

    $\displaystyle |a_n-A|< \frac{\epsilon}{2}$

    pick a natural number $\displaystyle N_2$ so for all n> $\displaystyle N_2$

    $\displaystyle |b_n-A|< \frac{\epsilon}{2}$

    let $\displaystyle N_3$ be the max of $\displaystyle N_1$ and $\displaystyle N_2$

    the go for it. for all n > $\displaystyle N_3$

    $\displaystyle |a_n+b_n -A-B|=|(a_n-A)+(b_n-B)|$


    use the triangle inequality and you have got it made
    Last edited by TheEmptySet; Mar 6th 2008 at 04:10 PM.
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