ε-close sequences

Let $(a_n{)}_{n=0}^{\mathrm{\infty }}$ and $(b_n{)}_{n=0}^{\mathrm{\infty }}$ be two sequences and let $\epsilon >0$ be a rational. $(a_n{)}_{n=0}^{\mathrm{\infty }}$ is said to be ε-close to $(b_n{)}_{n=0}^{\mathrm{\infty }}$ iff [...] for all $k\ge 0$.