I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct:
$$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j \rightarrow \infty, \text { such that } \varphi_{t_j} (x) \rightarrow y \text{ as } j \rightarrow \infty \}.$$
The context is ODEs, so $M$ is the phase space.
Given a point $x \in M$, its $\omega$-limit set with respect to a flow $\varphi_t$ is defined by $$ \omega(x)=\bigcap_{y \in \gamma(x)} \overline {\gamma^+(y)}, $$ where $$ \gamma(x)=\big\{\varphi_t(x): t \in \mathbb R\big\}\quad\text{and}\quad\gamma^+(y)=\big\{\varphi_t(y): t >0\big\}. $$ When $M$ is for example a smooth manifold (or simply $\mathbb R^n$), one can show that $$ \omega(x)=\big\{y\in M:\text{there exists a sequence $t_k \to +\infty$ such that $\varphi_{t_k}(x) \to y$ when $k \to \infty$}\big\}. $$