Suppose that lim sn = s, with s > 0. Prove that there exist a real number N such that sn> 0 for all n > N.
In the definition of sequence convergence, set $\displaystyle \varepsilon = \frac{s}{2}$.
Then $\displaystyle \left( {\exists N} \right)\left( {\forall n > N} \right)\left[ {\left| {s_n - s} \right| < \varepsilon } \right]\, \Rightarrow \,s_n > \frac{s}{2} > 0$