How to prove regular languages are closed by some operations?

I don’t how to prove these.

Show that the regular languages are closed under the following operations:

(a) $$\mathbb{DROPOUT}(L) = \{ xz \mid xyz \in L \text{ where }x,z \in \Sigma^*, y \in \Sigma \}.$$ Namely, $$\mathbb{DROPOUT}(L)$$ is the language containing all strings that can be obtained by removing one symbol from a string in $$L$$. For example, if $$L = \{012\}$$, then $$\mathbb{DROPOUT}(L) = \{12, 02, 01\}$$.

(b) $$\mathbb{INIT}(L) = \{ w\in \Sigma^+ \mid \text{ for some } x\in\Sigma^*,\ wx \in L\}.$$ For example, if $$L = \{01, 110\}$$, then $$\mathbb{INIT}(L) = \{0, 01, 1, 11, 110\}$$. (HINT: Start with a DFA $$A$$ for $$L$$ and describe how to construct an FA for $$\mathbb{INIT}(L)$$ using $$A$$. We assume that $$A$$ has no sink states.)