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Math Help - Analysis: Sequences

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    Analysis: Sequences

    Suppose lim Sn = S with S > 0.. Prove that there exists a real number N such that Sn > 0 for all natural numbers n.
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  2. #2
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    Quote Originally Posted by GoldendoodleMom View Post
    Suppose lim Sn = S with S > 0.. Prove that there exists a real number N such that Sn > 0 for all natural numbers n.
    \begin{gathered}<br />
  S > 0 \Rightarrow \quad \frac{S}<br />
{2} > 0 \hfill \\<br />
  \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \left| {s_n  - S} \right| < \frac{S}<br />
{2}} \right] \hfill \\<br />
   - \frac{S}<br />
{2} < s_n  - S < \frac{S}<br />
{2} \Rightarrow \quad 0 < \frac{S}<br />
{2} < s_n  < \frac{{3S}}<br />
{2} \hfill \\ <br />
\end{gathered}
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