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If $L$ is a regular language, how to prove $L_1 = \{ uv \mid u \in L, |v| =2 \}$ is also regular?

If $L$ is a regular language, prove that the language $L_1 = \{ uv \mid u \in L, |v| =2 \}$ is also regular. My idea: $L$ can be represented as a DFA and then you could add 2 consecutive ...
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Are supersets of non-regular languages also non-regular?

I have to proof that if $L_1 \subset L_2$ and $L_1$ is not regular then $L_2$ it not regular. This is my proof. Is it valid? Since $L_1$ is not regular, there does not exists a finite automata $M_1$ ...