$$x\sim \frac{x-np}{\sqrt{npq}} \overset{d}{\to} N(0,1)$$
I want to show that normalised binomial distribution converges in distribution to standard normal distribution.
Note that: Convergence in distribution mean
$F$ is cumulative distribution function
If $F_n(x) \to F(x)$ as $n \to \infty $ then $x_n \overset{d}{\to} x$
Also note that
$$M_{x_n}(t) \to M_x(t) \Rightarrow F_n(x) \to F(x) \Rightarrow x_n \overset{d}{\to} x$$
where $M$ is called as moment generating function.
so I need to show that
$$\lim_{n\to \infty} M_{x_n}(t)=M_x(t)$$
For binomial distribution, the moment generating function is $[pe^t +(1-p)]^n$
And for standard normal distribution, the moment generating function is $e^{t^2/2}$