I'm quite sure the following is true, but I'm not sure how to prove it.
Fix some $p\in(0,1)$. For any $p'\in[0,1]$, there exists sequences $\{x_n,y_n\}_{n=0}^\infty \subset (0,1)$ s.t. $$\lim_{n\to\infty} \frac{x_np}{x_n p + y_n(1-p) } =p'$$
Motivation for this question: in broad strokes, proving the above illustrates Sequential Equilibria and Perfect Bayesian Equilibria are equivalent in simple signaling games. (Apologies if this is still cryptic. A "proper" explanation would require a much, much, longer post.)