Prove every strongly connected tournament has a cycle of length k for k = 3, 4, … n where n is the number of vertices. Prove every strongly connected tournament has a cycle of length $k$ for $k = 3, 4,  … n$ where $n$ is the number of vertices.
I know that I need to prove this thm by induction but I am not sure how.
 A: I suppose you already know that a strongly connected tournament of order $n\gt1$ must contain a $3$-cycle. (If the tournament has no $3$-cycle, it's a transitive tournament, and not strongly connected.) For the induction step you have to show that, given a $k$-cycle (where $3\le k\lt n$), you can construct a $(k+1)$-cycle.
Consider a $k$-cycle $C=\{u_0,\dots,u_{k-1}\}$ with arcs $u_iu_{i+1}$ (addition modulo $k$).
Case I. For some vertex $v\notin C$ there is an arc from $C$ to $v$ and there is also an arc from $v$ to $C$.
Then for some $i\in\{0,\dots,k-1\}$ there are arcs $u_iv$ and $vu_{i+1}$. We get a cycle on the vertex set $C\cup\{v\}$, replacing the arc $u_iu_{i+1}$ with the arcs $u_iv$ and $vu_{i+1}$.
Case II. For each vertex $v\notin C$, either all arcs between $C$ and $v$ are directed from $C$ to $v$, or alse all arcs between $C$ and $v$ are directed from $v$ to $C$.
Let $V$ be the set of all vertices $v$ such that all arcs between $C$ and $v$ are directed from $C$ to $v$, and let $W$ be the set of all vertices $w$ such that all arcs between $C$ and $w$ are directed from $w$ to $C$.
Since the tournament is strongly connected (and $k\lt n$), the sets $V$ and $W$ are nonempty, and there is at least one arc $vw$ with $v\in V$ and $w\in W$.  Then $\{u_0,\dots,u_{k-2},v,w\}$ is a $(k+1)$-cycle.
A: You can find a very detailed proof in this paper page 15 theorem 4
