Let $t_n$ be a convergent sequence and suppose that $\lim(t_n)>a$. Prove there exists a number $N$ such that $n>N$ implies $t_n>a$.

Question

My start: Say $\lim(t_n)=b$ and let $\epsilon>0$. Then there exists $N$ such that $n>N$ implies $|t_n-b|<\epsilon$.

or should I say:

Let $a>0$. Then there exists $N$ such that $n>N$ implies $|t_n-b|<a$.

Am I going in the right direction? Any tips of where to go from here would be helpful.