Sports Competition

Let's say we halt the competition at an arbitrary point. Assume that no two teams have competed in the same number of matches.

The most matches a team can have played is $14$ (once each against every other team); the least is $0$ . There are exactly $15$ integers in the list $0,1,\cdots,14$ ; so one team (call it $T_0$ ) must have played no matches, one team has played one match, etc, and one team, $T_{14}$ , must have played all $14$ matches. But $T_{14}$ cannot have played $T_0$ ! Contradiction; so it is always possible to find two teams that have played the same number of matches.