In a sports tournament involving $N$ teams, each team plays every other team exactly once. At the end of every match, the winning team gets 1 point and the losing team gets 0 points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:
$$x _ { 1 } \geq x _ { 2 } \geq \cdots \geq x _ { N }$$
Prove that for any $1 \leq k \leq N$,
$$\frac { N - k } { 2 } \leq x _ { k } \leq N - \frac { k + 1 } { 2 } .$$