This is from "Differential Equations, Dynamical Systems & An Introduction to Chaos" 2nd Ed.

Proposition: "The $\displaystyle \alpha$-limit set and the $\displaystyle \omega$-limit set of a solution that is defined for all t$\displaystyle \in$R are closed, invariant sets."

The book doesn't provide a proof, and I was wondering how come this is true.