$ L_1^*∪L_2^*⊆(L_1∪L_2)^*$

I actually disproved the opposite proposition $ [(L_1∪L_2)^*⊆L_1^*∪L_2^*]$ and my intuition tells me that this is actually true… I tried to show that the combinations of words in the right group $ [(L_1∪L_2)^*]$ obviously contain all the words from the left group $ [L_1^*∪L_2^*]$ and it’s also very plain to see, but I have no idea how to formalize it.

Any idea how to formally prove it? Thanks.