# Positive Sequence with Positive Limit

• August 17th 2011, 03:00 PM
veronicak5678
Positive Sequence with Positive Limit
Suppose that sn is a sequence of positive numbers converging to a positive limit L. Show that there is a c > 0 such that sn > c for all n.

I can do this for sn < c using the limit definition, but I don't know how to show this.

• August 17th 2011, 03:27 PM
Plato
Re: Positive Sequence with Positive Limit
Quote:

Originally Posted by veronicak5678
Suppose that sn is a sequence of positive numbers converging to a positive limit L. Show that there is a c > 0 such that sn > c for all n.

I can do this for sn < c using the limit definition, but I don't know how to show this.

First of all either learn to use LaTeX or do not use special fonts.

Suppose that $(s_n)\to L>0$.
Using $\epsilon=\frac{L}{2}$ and the definition of sequence convergence prove that there is a p.i. N such that $n \geqslant N\, \Rightarrow \,s_n > \frac{L}{2}$.

Now let $c = \min \left\{ {\frac{L}{2},s_1 ,s_2 , \cdots ,s_{N - 1} } \right\} > 0$.

You are done if you fill in the details.