Every tournament has a Hamiltonian path

Idea. Insert the last player into an ordering of the others.

Base case. For $n=1$ there is nothing to prove.

Induction step. Take a tournament with $n+1$ players and remove a player $v$. By the induction hypothesis the other $n$ players can be ordered $P_1,\dots,P_n$ with $P_i$ beating $P_{i+1}$ for all $i$. We insert $v$.
- If $v$ beat $P_1$, put $v$ first.
- If $P_n$ beat $v$, put $v$ last.
- Otherwise $P_1$ beat $v$ and $v$ beat $P_n$. Let $j$ be the smallest index with $v$ beating $P_j$. It exists since $v$ beats $P_n$, and $j\ge2$. By minimality of $j$, $P_{j-1}$ beat $v$. Insert $v$ between $P_{j-1}$ and $P_j$.

In all cases we obtain the required ordering of all $n+1$ players.

Remark. Rédei's theorem (1934).

Lesson. Induction on the number of vertices: delete one, apply the hypothesis, and find a flexible place to reinsert.