Please consider the following excerpt.
I fail to understand how the hypothesis $\exists\alpha>0(A\backslash\{c\}\cap(c-\alpha,c+\alpha) = (S\backslash\{c\})\cap(c-\alpha,c+\alpha))$ seems to imply the clause in green.
Your guidance is appreciated as always.
In the hypothesis take intersection with $(c-\epsilon, c+\epsilon)$ on both sides. Note that $(c-a, c+a)\cap (c-b, c+b)=(c-a, c+a) $ if $a<b$. Finally, the fact that $(S\setminus \{c\}) \cap(c-\epsilon, c+\epsilon)$ is non-empty follows from the fact that $c$ is a cluster point.