I have little experience with math proofs and I would like to prove $A \cup (A\cap B)=A$, by showing that the left hand side is a subset of right hand side and vice versa.
Since $ A \subseteq A \space and \space A\cap B \subseteq A$
$\therefore \space lhs \subseteq rhs$
Given that $A \cup (A\cap B) \equiv A \cap (A \cup B)$
$A \subseteq A \space and \space A \subseteq (A\cup B) $
$\therefore \space rhs \subseteq lhs$
Does the above proposed solution suffice as a proof?