The problem asks to prove that there exists a connected graph in which the degrees of the nodes are $1,1,2,2, \ldots, 2000,2000.$ Label the nodes by $a_1, \ldots, a_{1000}, b_1 \ldots, b_{1000}$ and connect $a_1$ with all the $b_i$'s; connect $a_2$ with all the $b_i$'s, except for $b_{1000}$; connect $a_3$ with all the $b_i$'s, except for $b_{999}$ and $b_{1000}$; and so on. In this way, $a_i$ and $b_i$ have degree $1001-i,$ for all $1 \le i \le 1000$ and the resulted graph is connected. Note: The statement doesn't mention explicitly that the graph should be connected, but it's reasonable to assume that this kind of problems refer to road networks.