Let $S_{n}\sim\text{Bin}\left(n,p\left(n\right)\right)$ where $p\left(n\right)$ is the unique solution to the equation $\delta\left(p\left(n\right),n\right)=0$ with $\delta$ being continuous and bounded, strictly decreasing in $p$ with $\delta\left(0,n\right)>0$ and $\delta\left(1,n\right)<0$ for all $n\geq2$. We also have that $p\left(n\right)$ is strictly decreasing in $n$.
How to justify rigorously that there exists a unique (fixed) $p$ such that $\lim_{n\rightarrow\infty}\left(E\left[S_{n}\right]-n\cdot p\right)=0$, where $p$ is the unique solution to the equation $\lim_{n\rightarrow\infty}\delta\left(p\left(n\right),n\right)=0$?
It is not true without additional assumptions. Here is a counterexample.
For fixed $n$, Let $\delta(x,n)$ be the linear interpotation between $1$ and $0$ between $x=0$ and $x=\frac 12 + \frac 1n$, and the linear interpolation between $0$ and $-1$ between $x=\frac 12 + \frac 1n$ and $x=1$.
Then $\delta$ satisfies all the requirements and $p(n) = \frac 12 + \frac 1n$ is decreasing to $p=\frac 12$.
However,
$$E[S_n] - np = n (\frac 12 + \frac 1n) - n \frac 12 =1.$$