How to prove that $(a∪b)^*⊆b^*(ab^*)^*$? (formal languages)

Question

(a and b are two formal languages).
I tried to prove that using the fact that for languages $L_1, L_2$ we get: $(L_1∪L_2)^* =(L_1^* L_2^* )^*$ ,
but I got stuck here:

$(a∪b)^* = (a^* b^* )^*⊆b^* (a^* b^* )^*$

Answer 1

HINT: Prove by induction on $n$ that every string of length $n$ in $(a\cup b)^*$ is in $b^*(ab^*)^*$; use the fact that if $x$ is a string of length $n+1$, then there is a string $y$ of length $n$ such that either $x=ay$ or $x=by$. Treat the two cases separately. Essentially the induction step amounts to showing that

$$(a\cup b)b^*(ab^*)^*=ab^*(ab^*)^*\cup bb^*(ab^*)^*\subseteq b^*(ab^*)^*\;.$$