# $\omega$-limit set for $X \supset Y$, $X$ bounded

I'm working through a problem, and it states "Show that if $$X$$ is bounded and $$X \supset Y$$, then $$\omega(X) \supset \omega(Y)$$. Deduce that if $$Y$$ is an absorbing set then $$\omega(X) = \omega(Y)$$." (In this case, we are assuming $$S(t)$$ is a semigroup of solution operators form a $$C^0$$ semigroup, i.e. that are NOT linear in space.)

First question: if we use the definition $$\omega(X) = \{z : \exists t_n \to \infty, x_n \in X, S(t_n)x_n \to z\}$$, it seems trivial that $$\omega(Y) \subset \omega(X)$$. I don't know why it is required that $$X$$ be bounded. Shouldn't it simply follow from the definition?

Additionally, it says deduce that if $$Y$$ is an absorbing set, then the $$\omega$$-limit sets are equal. If $$Y$$ is absorbing, then so is $$X$$. We can maybe use the other definition here, where $$\omega(X) = \bigcap\limits_{t \geq 0} \overline{\bigcup\limits_{s\geq t} S(s)X}$$, which since $$Y$$ is absorbing we can rewrite $$\omega(X) = \bigcap\limits_{t \geq 0} \overline{\Big(\bigcup\limits_{t_0 \geq s \geq t} S(s)X \cup Y\Big)}$$ and $$\omega(Y) = \bigcap\limits_{t \geq 0} \overline{\Big(\bigcup\limits_{t_0 \geq s \geq t} S(s)Y \cup Y\Big)}$$, for an appropriately chosen $$t_0$$. I don't see how we could use this to make a simple deduction, so I am not sure what I am missing. Any help would be much appreciated.