Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
Go forward while you can, but if your strength fails you, sit down near the road and gaze without anger or envy at those who pass by. They don't have far to go, either.