I'm trying to learn automata theory on my own and I am running into an issue with the second part of the question:
We say B is transitive if $BB\subseteq B$ and reflexive if $\epsilon \in B$
Show that A* is a reflexive and transitive set containing A and if B is any other reflexive and transitive set containing A, then $A^*\subseteq B$.
I've shown that Kleene star satisfies these two conditions. I've tried partitioning B into two sets with $A = B \cup C $ and trying a constructive proof but this hasn't led any where. I also am considering a proof by contradition but don't know where to start.
Can you help me with a hint on how to approach this problem?