Strategy Stealing is another classic example that applies to a number of turn-based games. It shows that either the first player always wins or that the game will end in a tie, assuming perfect play from both sides. The proofs never actually exhibit the strategies in question.
For example take Tic Tac Toe (on an arbitrarily large board of size $n\times n$). Suppose player 2 has a winning strategy $S$, regardless of player 1's first move. Then we make a number of observations:
1) Regardless of where player 1 plays the first $X$, player 2 supposidly has a winning strategy, which is a function of the position of the first $X$.
2) There is never a disadvantage to having one of your pieces already on the board, meaning that if player 1 already has an $X$ on a given square, then that cannot the worse than not having an $X$ on that square.
3) By 1), player 1 can adopt player 2's strategy by randomly placing an $X$, and then after player 2 responds with their strategy $S$, player 1 applies $S$ to player 2's response, with $X$ and $O$ switched. If $S$ ever calls to play on the first $X$ that player 1 had to place, then player 1 can make a random move by 2).
So player 2 could not possibly have a winning strategy $S$, which means either player 1 always wins or the game always ends in a tie.