From below, it is the proof of the second proposition.
For the first part of the proof, I don't understand the statement 'if $x$ is in $A \setminus\{c\}$, then $x$ is in $S \setminus \{c\}$, and hence $f|_A(x) \rightarrow L$ as $x \rightarrow c$.'.
Of course, "if $x$ is in $A \setminus\{c\}$, then $x$ is in $S \setminus \{c\}$" is true, but we only know $x\in S \setminus \{c\}$, but do not know "$x \in A \setminus \{c\}$", then how does it help prove "$f|_A(x) \rightarrow L$ as $x \rightarrow c$".
Of course if $A\subseteq S$, anything in $A$ must be contained in $S$, so $x\in A-\{c\}$ implies that $x\in S-\{c\}$. And out goal is to prove that, whether or not, we have something like $\big|f|_{A}(x)-L\big|<\epsilon$ for all $x\in A-\{c\}$.
So we must first assume that $x\in A-\{c\}$, and then we judge whether or not we have the conclusion that $\big|f|_{A}(x)-L\big|<\epsilon$. And this is successfully done because we have the premise that if $x\in S-\{c\}$ then $|f(x)-L|<\epsilon$.