Induction Question. Suppose there are n teams in a football league

Question

how can i prove this

Let n > 1 be an integer. Suppose there are n teams in a football league and every two teams have played against each other exactly once with no ties. Prove that it is possible to number the teams 1 to n such that at the end of the season team i beats team i + 1 for i = 1,...,n − 1.

I proved this by for example giving n 3 or 4, but what exactly this question want ?

Answer 1

Define the $n$ teams as $T_i, 1\le i\le n$, and the win/lose relation as $\gt$ where $T_i$ beats $T_j$ iff $i\lt j$, so for example $T_2$ beats $T_3$. This means $T_1$ wins $n-1$ games, $T_2$ wins $n-2$ games and so on, with $T_n$ winning zero games. As each team has won a different number of games, your question has been answered.

Answer 2

Suppose the first $n-1$ have been arranged as asked. Try to show that it is possible either
1) to slot the nth team between two teams, or
2) that the nth team beat the first of the $n-1$, or
3) that the nth team lost to the last of the $n-1$.