Prove or disprove the following proposition $L_1^*∪L_2^*⊆(L_1∪L_2)^*$
This is true. 
You only have to prove the following staments:


if $A\subseteq B$ then $A^{*} \subseteq B^{*}$,
If $A\subseteq C$ and $B\subseteq C$ then $A \cup B \subseteq C$.


We know $L_1 \subseteq L_1 \cup L_2$ for 1. we can conclude $L_1^{*} \subseteq (L_1 \cup L_2)^{*}$, analogously $L_2^{*} \subseteq (L_1 \cup L_2)^{*}$. For 2. we can conclude $L_1^{*} \cup L_2^{*} \subseteq (L_1 \cup L_2)^{*}$.