The row ganadora

1. Try building the row little by little.
2. If you already have a valid row for k players, try inserting the new player in the appropriate place.
3. Go down the line until you find the first player the new player has beaten.

Answer: you are right. That line always exists. We are going to build it by incorporating the eight players one by one. With only one player, the line already meets the condition: there is no one behind him. Now suppose we have already got a valid row with $k$ players, where $1\le k<8$: $$
P_1\to P_2\to\cdots\to P_k,

$$ and the arrow means “won.” We want to add a new player $X$. If $X$ beat $P_1$, we put it at the beginning: $$

X\to P_1\to P_2\to\cdots\to P_k.

$$ If $P_k$ beat $X$, we put it last: $$

P_1\to P_2\to\cdots\to P_k\to X.

$$ The case remains in which it cannot go to the beginning or the end. So $P_1$ beat $X$, but $X$ beat $P_k$. We go down the row from the beginning and look for the first player that $X$ has beaten. That player exists, because $X$ beat $P_k$. Call it $P_{i+1}$. Since he is the first one $X$ beat, the previous player, $P_i$, had to beat $X$. So we have: $$

P_i\to X

$$ and $$

X\to P_{i+1}.

$$ We insert $X$ between them: $$

P_1\to\cdots\to P_i\to X\to P_{i+1}\to\cdots\to P_k.
$$ The row is still valid, because we have only replaced the $P_i\to P_{i+1}$ link with two valid links: $P_i\to X\to P_{i+1}$. This procedure allows you to go from a valid row of $k$ players to a valid row of $k+1$ players. We start with 1 player and repeat the process until we reach 8. Thus we obtain a valid row with the eight chess players. Furthermore, the reasoning does not use anything special about the number 8: it works the same for any number of players. In every tournament without ties there is a row in which each player has beaten the next.