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Math Help - {a_n}\to A iff. {a_n-A}\to 0

  1. #1
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    {a_n}\to A iff. {a_n-A}\to 0

    Show that \{a_n\}_{n=1}^{\infty}\to A iff. \{a_n-A\}_{n=1}^{\infty}\to 0

    (i) \Rightarrow
    Suppose \{a_n\}_{n=1}^{\infty}\to A. Let \epsilon>0 \ \text{and} \ \exists N\in\mathbb{N} such that |a_n-A|<\epsilon.

    \Rightarrow |a_n-A|=|(a_n-A)-0|<\epsilon
    \Rightarrow \{a_n-A\}_{n=1}^{\infty}\to 0

    (ii) \Leftarrow
    Now, suppose \{a_n-A\}\to 0. Let \epsilon>0 \ \text{and} \ \exists N\in\mathbb{N} such that |(a_n-A)-0|<\epsilon.

    \Rightarrow|(a_n-A)-0|=|a_n-A|<\epsilon
    \Rightarrow \{a_n\}_{n-1}^{\infty}\to A

    Is this all that needs to be done?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: {a_n}\to A iff. {a_n-A}\to 0

    Quote Originally Posted by dwsmith View Post
    Is this all that needs to be done?
    Yes, your proof is correct.
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