Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that the central limit theorem holds for this particular sequence. That means
$$\frac{\sum_{i=1}^n(X_i-\mu)}{\sigma \sqrt{n}} \xrightarrow{d} \mathcal{N}(0,1)$$
where $\mathbb E(X_1)=2\varepsilon$ and $\mathbb V(X_1)=1-4\varepsilon^2$.
Or in terms of the characteristic function:
$$\varphi_\frac{\sum_{i=1}^n(X_i-\mu)}{\sigma \sqrt{n}} (t) \to e^{-\frac{t^2}{2}}$$
I try to figure out the characteristic function:
$$\varphi_\frac{\sum_{i=1}^n(X_i-\mu)}{\sigma \sqrt{n}}(t)=\left(\varphi_\frac{X_i-\mu}{\sigma}\left(\frac{t}{\sqrt{n}}\right)\right)^n$$
where:
$$\varphi_\frac{X_i-\mu}{\sigma}(s)=\left(\frac{1}{2}-\varepsilon\right)\exp\left(is\frac{-1-\mu}{\sigma}\right)+\left(\frac{1}{2}+\varepsilon\right)\exp\left(is\frac{1-\mu}{\sigma}\right)$$
The expression seems to be unmanageable.
What have I done wrong? How can we fix it?