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Math Help - how to open an innequality

  1. #1
    MHF Contributor
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    how to open an innequality

    how they get from this expression

    |An+1 - An - L|<e

    to

    L-e<An+1 - An<L+e

    cant understand this transition
    ??
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by transgalactic View Post
    how they get from this expression

    |An+1 - An - L|<e

    to

    L-e<An+1 - An<L+e

    cant understand this transition
    ??
    We know that if \left|\text{whatever}\right|<\varepsilon then it is also true (by definition) that -\varepsilon<\text{whatever}<\varepsilon. So applying this concept to your inequality we get \left|A_{n+1}-A_n-L\right|<\varepsilon\implies -\varepsilon<A_{n+1}-A_n-L<\varepsilon. Adding L to the inequality gives L-\varepsilon<A_{n+1}-A_n<L+\varepsilon
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