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Math Help - Sequences

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    Sequences

    Suppose that lim sn = s, with s > 0. Prove that there exist a real number N such that sn> 0 for all n > N.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by bearej50 View Post
    Suppose that lim sn = s, with s > 0. Prove that there exist a real number N such that sn> 0 for all n > N.
    a proof by contradiction should work. what if s_n \le 0 for all n? consider the cases s_n = 0 for all n and s_n < 0 for all n separately
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    In the definition of sequence convergence, set \varepsilon  = \frac{s}{2}.
    Then \left( {\exists N} \right)\left( {\forall n > N} \right)\left[ {\left| {s_n  - s} \right| < \varepsilon } \right]\, \Rightarrow \,s_n  > \frac{s}{2} > 0
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