It is required to obtain the limiting probability
$$\lim_{n\rightarrow \infty}P\left (\frac{X_1-X_2+\cdots+X_{2n-1}-X_{2n}}{\sqrt{n}} \leq x \right )$$
The random variables are IID with variance $1.$
My approach
The variance of the $X_1-X_2+\cdots+X_{2n-1}-X_{2n}$ will be $2n$. One $\sqrt{n}$ is already in the denominator. Hence, we would divide the both sides by $\sqrt{2}$. So, the Variable $\frac{X_1-X_2+\cdots+X_{2n-1}-X_{2n}}{\sqrt{2n}}$ will be a standard Normal. But, how to proceed from here.
I don't have answers available. That's why I am getting difficulty in this problems.
Any help?