I understand the example proof he gives up to a point. I don't know how he changes the terms and support on his integral listed below. I also don't understand how it becomes a sum from a difference. Here is an excerpt. Excerpt
Given that $ R=X_{(n)}-X_{(1)}\le a $,
$$ P(R\le a)=n\int_{-\infty}^\infty[F(x_1+a)-F(x_1)]^{n-1}f(x_1)dx_1 $$
Equation (6.7) can be evaluated explicitly only in a few special cases. One such case is when the Xi’s are all uniformly distributed on (0, 1). In this case, we obtain, from Equation (6.7), that for 0 < a < 1,
$$ P(R \lt a)=n\int_{0}^1[F(x_1 + a)-F(x_1)]^{n-1}f(x_1)dx_1=n\int_{0}^{1-a}a^{n-1}dx_1+n\int_{1-a}^1(1-x_1)^{n-1}dx_1 $$