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Math Help - Limit Proof

  1. #1
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    Limit Proof

    Prove lim(x sub n) = 0 if and only if lim(|x sub n|)= 0.

    I am not sure how to even start this or what property of limits apply here.
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  2. #2
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    Quote Originally Posted by hayter221 View Post
    Prove lim(x sub n) = 0 if and only if lim(|x sub n|)= 0.
    Surely you know this: \left| {\left| {x_n } \right| - 0} \right| = \left| {\left| {x_n } \right|} \right| = \left| {x_n } \right| = \left| {x_n  - 0} \right|
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  3. #3
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    Can you show me how to incorporate this into the proof? I have that for every epsilon>0 there exists a natural number K(epsilon) such that for all n>or equal to K(epsilon) the tems x sub n satisfy |x sub n -0|< epsilon. From what you said |x sub n| = |x sub n -0| but I am not sure how this leads to lim(|x sub n|) = 0 from lim(x sub n)=0 or vice versa.
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  4. #4
    MHF Contributor kalagota's Avatar
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    well, by assumption, \lim x_n = 0 iff for every \varepsilon>0 there exists a natural number K such that for all n \geq K we have |x_n - 0| < \varpesilon..

    and from plato's post, the last in the equality is less than epsilon..

    in the same manner, \lim |x_n| = 0 iff for every \varepsilon>0 there exists a natural number K such that for all n \geq K we have ||x_n| - 0| < \varpesilon..
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