Michael Sipser Edition 3 Exercise 1 Question 63 (Page No. 92)

Question

1. Let $A$ be an infinite regular language. Prove that $A$ can be split into two infinite disjoint regular subsets.
2. Let $B$ and $D$ be two languages. Write $B\subseteqq D$ if $B\subseteq D$ and $D$ contains infinitely many strings that are not in $B.$ Show that if $B$ and $D$ are two regular languages where $B\subseteqq D,$ then we can find a regular language $C$ where $B\subseteqq C\subseteqq D.$