How does $\lim\limits_{n \to \infty} \dfrac{p[s + (1 - s)c^n]}{p[ s + (1 - s)c^n] + (1 - p)[s + (1 - s)w^n]} = p?$

−0

So if $n \to \infty$ , then all terms above containing $c^{n} \to 0$ .

This is false. It's true that as $n \to \infty$ , $c^{n} \to 0$ (for $0 < c < 1$ ). But a term like $\frac{s}{c^{n}}$ will diverge instead of converging to $0$ .

You haven't done yourself any favors by dividing everything by $c^{n}$ . Just use the fact that $c^{n} \to 0$ (and similarly $w^{n} \to 0$ ) with the original expression, go slowly, and you should get the right answer.