In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums.
The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0: This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense, to show that if A + B = 0 then A = B = 0.
In geometric topology the addition used in the swindle is usually the connected sum of knots or manifolds.
Example: The oriented n-manifolds have an addition operation given by connected sum, with 0 the n-sphere.
This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows.