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    real analysis test question

    Suppose that lim s(sub n) =s, with s >0. Prove that there exists an N in R (real numbers) such that s(sub n) >0 for all n in N.
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    Hello,
    Quote Originally Posted by trojanlaxx223 View Post
    Suppose that lim s(sub n) =s, with s >0. Prove that there exists an N in R (real numbers) such that s(sub n) >0 for all n in N.
    \lim s_n=s means that :
    \forall \epsilon>0, \exists N \in \mathbb{N}, \forall n\geq N, |s_n-s|< \epsilon

    Now take \epsilon such that s-\epsilon is positive. You can always find one, such as \epsilon=\frac s2

    From the inequality |s_n-s|<\epsilon, we can say that s-\epsilon<s_n<s+\epsilon
    Since we defined \epsilon such that s-\epsilon>0, it's easy to conclude.
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