Given the alphabet $\Sigma$ and the languages $L_1$ and $L_2$ over the given alphabet I was asked to prove/disprove the next statement: $$ (L_1 \cup L_2)^* = L_1^* \circ L_2^*$$
I've run a few examples and I'm pretty sure that the statement is correct. I'd like some help on hints/clues on how to prove it :) I know that I need to prove that both of the sets are equal but I'm struggling with the formality involved.
The simplest counter-example is $L_1=\{0\}$ and $L_2=\{1\}$.