Question Let $(X_n)_{n\geq 1}$ be a sequence of arbitrary binomial random variables such that $EX_n\to \infty$ and $\text{Var}(X_n)/EX_n^2\to 0$ as $n\to \infty$. Then show that $$ Z_n=\frac{X_n-EX_n}{\sqrt{\text{Var}(X_n)}}\stackrel{\text{d}}{\to } N(0,1). $$
My attempt
The idea of the proof is to find the characteristic function $\varphi_{n}$ of $Z_n$ and show that $\varphi_{n}(t)\to \exp(-t^2/2)$.
To this end, let $X_n\sim \text{Bin}(m_n,p_n)$. Since $\varphi_{X_n}(t)=(1-p_n+p_ne^{it})^{m_n}$ we have that $$ \varphi_{n}(t)=\varphi_{Z_n}(t)=\exp\left( \frac{-itm_np_n}{\sqrt{m_np_n(1-p_n)}} \right) \left(1-p_n+p_n\exp\left(\frac{it}{\sqrt{m_np_n(1-p_n)}}\right)\right)^{m_n}. $$ At this point I tried to taylor expand the exponential terms, mimic the proof of classical clt and use the fact that if $c_j\to 0$, $a_j\to \infty$, $a_jc_j\to \lambda$, then $(1+c_j)^{a_j}\to e^{\lambda}$ as $j\to \infty$. Doing so yields for example that $$ 1-p_n+p_n\exp\left(\frac{it}{\sqrt{m_np_n(1-p_n)}}\right)=1+p_n\frac{it}{\sqrt{m_np_n(1-p_n)}}-p_n\frac{t^2}{2(m_np_n(1-p_n))}+\dotsb $$ and $$ \exp\left(-\frac{itm_np_n}{\sqrt{m_np_n(1-p_n)}}\right)=1-\frac{itm_np_n}{\sqrt{m_np_n(1-p_n)}}-\frac{t^2m_n^2p_n^2}{2m_np_n(1-p_n)}+\dotsb. $$ In the first of these we have something that resembles the term $-t^2/2$ but I can't proceed from here.
Any help is appreciated.