Limit distribution of a normalised sum of differences of pairs

Let $X_1,X_2,…$ be a sequence of i.i.d. random variables with variance $1$. Then $\lim_{n\to\infty}P\left(\dfrac{(X_1-X_2)+(X_3-X_4)+\cdots+(X_{2n-1}-X_{2n})}{\sqrt n}\le x\right)$ is equal to