For any language $L_1$ and $L_2$ if $L_1 \subseteq L_2$, then ${L_1}^* \subseteq {L_2}^*$

Question

I am tempted to say no, since you can not say that an infinite set is a subset if an an infinite set. Unless you are talking about something like the natural numbers is a subset of the real numbers but here they are talking about any language. Any pointers?

Answer 1

This is true. Assume $L_1 \subseteq L_2$. We then have for all $n \in \mathbb{N}_0$ that $L_1^n \subseteq L_2^n$. This implies that

$\bigcup_{n \in \mathbb{N}_0} L_1^n \subseteq \bigcup_{n \in \mathbb{N}_0} L_2^n $

which is just the definition of $L_1^*\subseteq L_2^*$.