Definition 4.3.2 (Eventually). Let $X$ be a set, $\net{x}\subset X$, and $E \subset X$, then $\net{x}$ is eventually in $E$ if there exists $\alpha_{0} \in A$ such that $x_{\alpha} \in E$ for all $\alpha \gtrsim \alpha_{0}$.
Definition 4.3.2 (Eventually). Let $X$ be a set, $\net{x}\subset X$, and $E \subset X$, then $\net{x}$ is eventually in $E$ if there exists $\alpha_{0} \in A$ such that $x_{\alpha} \in E$ for all $\alpha \gtrsim \alpha_{0}$.