Limit points

Let $(a_n{)}_{n=m}^{\mathrm{\infty }}$ be a sequence of real numbers, let x be a real number, and let $ϵ>0$ be a real number. We say that $x$ is ε-adherent to $(a_n{)}_{n=m}^{\mathrm{\infty }}$ iff there exists an $n\ge m$ such that [...]. We say that $x$ is continually ε-adherent to $(a_n{)}_{n=m}^{\mathrm{\infty }}$ if it is ε-adherent to [...] for every $N\ge m$. We say that $x$ is a limit point or adherent point of $(a_n{)}_{n=m}^{\mathrm{\infty }}$ if it is continually ε-adherent to [...] for every [...].