Relabelling players in a tournament BdMO 2014

$n$ players take part in a chess tournament where each player plays with all others only once and the only outcomes of the games are win and loss.Prove that it is possible,after the tournament has ended,to label the players such that Player 1 has defeated P-2,P-2 has defeated P-3,P-3 has defeated P-4. . . .P-(n-1) has defeated P-n (here,P-i means the i-th player).

We draw n points(indicating n players) such that no 3 are collinear.Then each edge between two vertices indicate a game between two players.Then I am lost,mainly because I am not familiar with graph theory and also because I can't find any other way to express what the question wants mathematically.Here are two questions:
1)Can this be proved using graph theory,If so,how?
2)Can this be proved without graph theory?If so,how?
 A: This can certainly be proven using graph theory. In fact, this is a classical result related to digraphs called (appropriately) tournaments. Your problem is equivalent to that of finding a Hamiltonian path in the tournament. This statement is proven right in the Wikipedia article on tournaments which I linked to above. For completeness, let me reproduce the argument.
We prove the statement by induction on $n$, the number of vertices/players. The base case of $2$ players is obvious. Suppose that any tournament on $n$ players has a Hamiltonian path. 
Now consider a tournament on $n+1$ players. Remove a single vertex $v$ of this tournament and consider the remaining $n$ player tournament. By hypothesis, there exists Hamiltonian path, say
$$v_1\rightarrow v_2\rightarrow \cdots\rightarrow v_n$$
Let $i\in \{1,\ \cdots,\ n\}$ be maximal such that each $v_j$ for $j\le i$ has a directed edge from $v_j$ to $v$. Then 
$$v_1 \rightarrow \cdots \rightarrow v_i \rightarrow v \rightarrow v_{i+1} \rightarrow \cdots \rightarrow v_n$$
is a Hamiltonian path on the $n+1$ player tournament.
