Suppose that lim s(sub n) =s, with s >0. Prove that there exists an N in R (real numbers) such that s(sub n) >0 for all n in N.
Hello,
$\displaystyle \lim s_n=s$ means that :
$\displaystyle \forall \epsilon>0, \exists N \in \mathbb{N}, \forall n\geq N, |s_n-s|< \epsilon$
Now take $\displaystyle \epsilon$ such that $\displaystyle s-\epsilon$ is positive. You can always find one, such as $\displaystyle \epsilon=\frac s2$
From the inequality $\displaystyle |s_n-s|<\epsilon$, we can say that $\displaystyle s-\epsilon<s_n<s+\epsilon$
Since we defined $\displaystyle \epsilon$ such that $\displaystyle s-\epsilon>0$, it's easy to conclude.