ILL 1972 Problem 18

We have $p$ players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers $s_1 \le s_2 \le s_3 \le\cdots\le s_p$ is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if $$(i) \sum_{i=1}^{p} s_i =\frac{1}{2}p(p-1)$$ $$\text{and}$$ $$(ii)\text{ for all }k < p, \sum_{i=1}^{k} s_i \ge \frac{1}{2} k(k - 1).$$