proving L1* ∪ L2* ⊆ (L1∪L2)*
No. Your proof of $A\iff B$ is to claim that

$$A\iff \text{something} \iff \text{something else} \iff A \iff B\,.$$

The first two stages are irrelevant, since you get back to&nbsp;$A$. The final stage just asserts as true the exact thing you're supposed to be proving.

Try instead to prove that every string in the first language is in the second.