This is the 3rd question from the 2nd chapter of the book "A Walk Through Combinatorics - Miklos Bona"
The question is as follows -
At a tennis tournament, there were $2^n$ participants, and any two of them played against each other exactly one time. Prove that we can find $n+1$ players that can form a line in which everybody has defeated all the players who are behind him in the line.
The proof stated is as follows -
"Induction on $n$. For $n = 1$, the statement is trivially true. Now assume the statement is true for n and prove it for $n + 1$. The winner $X$ of a tournament with $2^{n+1}$ participants must have won at least $2^n$ games (why?). Take $X$, and $2^n$ people he defeated. By the induction hypothesis, we can find $n + 1$ people among the $2^n$ people defeated by $X$ who can form a line with the required property. Then we put $X$ to the front of this line and we have obtained a line of length $n + 2$ that has the required property."
My problem with this -
The winner in a tournament with $2^{n+1}$ players has at least $2^n$ as proven here by @Martund -
Number of winners in a tournament
But the winner $X$ may have more than $2^n$ wins, in which case we may not find a set of $2^n$ players who lost to $X$ for us to apply induction and extract the line of $n$ players? How do we fix this?
How many total wins must exist in the tournament? The way to have the "winningest" player have the least number of wins is if each player has the same number of wins.
Can you flesh fill in the details to answer your own question?
Is your statement correct: "But the winner $X$ may have more than $2n$ wins, in which case we may not find a set of $2n$ players who lost to $X$"?