(Possibly Easy) Formal Language Question
The reasoning you give is not formal enough. For the union case, you're just restating the theorem "the union of regular sets is regular since the union of regular sets is regular"; for the difference case, your assertion "a subset of a regular language is regular" is false ($\Sigma^*$ is regular and every language is a subset of $\Sigma^*$).

Regular languages are defined in terms of finite automata. The general approach to a proof is that you look at the definitions of what is given (L1 and L2 are regular, so there are automata...) and show a logical reasoning from that to the definition of what you want to prove (there is an automaton ... therefore L3 is regular).