Must the champion of an entire tournament beat the champion of a possible tournament among other players?
Here is a proof by induction. The base case, $n = 2$, is clear.

Now consider a tournament in which $p_1$ wins. Suppose first that the first competition doesn't involve $p_1$. Let the new set of players be $p_1,P$. By induction, we know that $p_1$ must beat one of the possible winners of the tournament in $P$, say $p_i$. Since the first competition didn't involve $p_1$, $p_i$ can also win the tournament on $p_2,\ldots,p_n$, completing the proof in this case.

Suppose next that the first competition has $p_1$ beating $p_2$. Applying the induction hypothesis, we know that $p_1$ must beat one of the possible winners of the tournament in $p_3,\ldots,p_n$, say $p_i$. If $p_i$ beats $p_2$, then $p_i$ can also win the tournament on $p_2,\ldots,p_n$, and we're done. If $p_2$ beat $p_i$, then $p_2$ can win the tournament on $p_2,\ldots,p_n$, and so we're again done (since in the first competition, $p_1$ beat $p_2$).