Redei's theorem: every nonempty tournament has a directed Hamilton path

If $n=1$, the one-vertex directed path contains every vertex.

Assume every tournament on $n-1$ vertices has a directed Hamilton path, where $n>1$. Remove a vertex $x$. The remaining tournament has, by the induction hypothesis, a directed Hamilton path $v_1,v_2,\dots ,v_{n-1}$.

If $x{v}_1$ is an arc, prepend $x$. If no arc $x{v}_j$ exists, [F1] gives $v_{j}x$ for every $j$, so append $x$. Otherwise let $j$ be the least index with arc $x{v}_j$; then $j>1$ and minimality plus [F1] gives the arc $v_{j-1}x$. Insert $x$ between $v_{j-1}$ and $v_j$.

In every case the resulting list follows directed arcs and contains each vertex of $T$ exactly once, so it is a directed Hamilton path.

The base case and induction step prove Redei's theorem for every nonempty finite tournament.