Round Robin Tournaments

In round-robin tournament scheduling, we wish to assign a home team and an away team for each game so that each of n teams, where n is odd, plays an equal number of home games and away games.

Show that if, when i+j is odd, we assign the smaller of i and j as the home team, whereas if i+j is even, we assign the larger of i and j as the home team, then each team plays an equal number of home and away games.