# Analysis: Sequences

• November 16th 2008, 03:36 PM
GoldendoodleMom
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.
• November 16th 2008, 03:43 PM
Plato
Quote:

Originally Posted by GoldendoodleMom
Suppose lim Sn = S with S > 0.. Prove that there exists a real number N such that Sn > 0 for all natural numbers n.

$\begin{gathered}
S > 0 \Rightarrow \quad \frac{S}
{2} > 0 \hfill \\
\left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \left| {s_n - S} \right| < \frac{S}
{2}} \right] \hfill \\
- \frac{S}
{2} < s_n - S < \frac{S}
{2} \Rightarrow \quad 0 < \frac{S}
{2} < s_n < \frac{{3S}}
{2} \hfill \\
\end{gathered}$