Eventually ε-close sequences

Let $(a_n{)}_{n=0}^{\mathrm{\infty }}$ and $(b_n{)}_{n=0}^{\mathrm{\infty }}$ be two sequences of rational numbers and let $\epsilon >0$ be a rational. The sequences are said to be eventually ε-close iff there exists an integer $N\ge 0$ such that [...] are ε-close.