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Math Help - 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 N_1 so for all n > N_1

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

    pick a natural number N_2 so for all n> N_2

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

    let N_3 be the max of N_1 and N_2

    the go for it. for all n > N_3

    |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; March 6th 2008 at 04:10 PM.
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