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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- Nov 16th 2008, 02:36 PMGoldendoodleMomAnalysis: 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.

- Nov 16th 2008, 02:43 PMPlato
$\displaystyle \begin{gathered}

S > 0 \Rightarrow \quad \frac{S}

{2} > 0 \hfill \\

\left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \left| {s_n - S} \right| < \frac{S}

{2}} \right] \hfill \\

- \frac{S}

{2} < s_n - S < \frac{S}

{2} \Rightarrow \quad 0 < \frac{S}

{2} < s_n < \frac{{3S}}

{2} \hfill \\

\end{gathered} $