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.
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 $\displaystyle (s_n)\to L>0$.
Using $\displaystyle \epsilon=\frac{L}{2} $ and the definition of sequence convergence prove that there is a p.i. N such that $\displaystyle n \geqslant N\, \Rightarrow \,s_n > \frac{L}{2}$.
Now let $\displaystyle c = \min \left\{ {\frac{L}{2},s_1 ,s_2 , \cdots ,s_{N - 1} } \right\} > 0$.
You are done if you fill in the details.