The basin of attraction $B(x_0)$ of a sink $x_0$ of a dynamical system governed by $x'=f(x)$ (with $x\in \mathbb{R}^n$) is defined to be the set of points which, for any neighborhood $U$ of $x_0$, eventually end up in $U$. That is:
$$B(x_0)=\{ y\in \mathbb{R}^n \,|\, \forall \text{ open neighborhood } U \text{ of } x_0,\, \exists t'\in\mathbb{R}^+:\,(t\ge t'\Rightarrow\phi(t,y)\in U) \}$$
In other words:
$$B(x_0)=\{ y\in \mathbb{R}^n : \forall \text{ open neighborhood } U \text{ of } x_0,\, \lim_{t\rightarrow+\infty} \phi(t,y)=x_0\}$$
Is this equivalent to
$$B(x_0)=\{ y\in \mathbb{R}^n : \omega(y)={x_0}\}$$ (where $\omega(y)$ is the omega-limit of $y$)? I think it is, since: there exists a sequence $(t_k)$ of positive times which can be constructed such that $(t_k)$ goes to $\infty$ and $\lim_{t_k\rightarrow+\infty }\phi(t,y)=x_0$ (using the fact that $\lim_{t\rightarrow+\infty} \phi(t,y)=x_0$). Reciprocally, seems clear to me that the existence of a sequence of positive times such that $\lim_{t_k\rightarrow+\infty }\phi(t,y)=x_0$ implies that $\lim_{t\rightarrow+\infty} \phi(t,y)=x_0$.
However, I read that $\lim_{t\rightarrow+\infty} \phi(t,y)=x_0$ is stronger than the existence of a sequence such that $\lim_{t_k\rightarrow+\infty }\phi(t,y)=x_0$, What is it, after all?
