Let us consider the products in pairs, starting with $x_1x_3 + x_2x_4$. This relates to four consecutive terms. Setting $c = \max\{x_1, x_4\}$ gives $x_1x_3 \le c x_3$ and $x_2x_4 \le c x_2$. Moreover, the assumptions imply $c + x_2 + x_3 \le 2$ and so
$$x_1x_3 + x_2x_4 \le c(x_2 + x_3) \le \left(\frac{c + x_2 + x_3}{2}\right)^2 \le 1$$
using AM-GM in the middle. More generally, for $i = 1, 2, \dots, 1009$ the same logic implies $x_{2i-1}x_{2i+1} + x_{2i}x_{2i+2} \le 1$. Adding these inequalities gives the required result.