Suppose we have a language $A$. I want to prove that $AA^*$ is commutative. I know that this expression equals $A^+$, but I’m not sure how to go about a proof yet. This is my attempt so far.

If $A$ is a language, then

begin{align*}

AA^* & = A(A^0cup A^1cupcdots) tag{Definition of $A^*$}\

& = A^1cup A^2cdots tag{Distributive law} \

& = (A^0cup A^1cupcdots)A tag{Distributive law} \

& = A^*A.

end{align*}

Is this enough? I’m thinking that in order for it to be a proper proof, I have to show that $AA^*$ and $A^*A$ are subsets of each other. That is, take an element $s = xy$ where $xin A$ and $yin A^*$ and show that $xin A^*$ and $yin A$ (and vice versa). However, I get stuck very early when doing that method.