Eilenberg–Mazur swindle (nonfiction)

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.

In geometric topology it was introduced by Mazur (1959, 1961) and is often called the Mazur swindle.

In algebra it was introduced by Samuel Eilenberg and is known as the Eilenberg swindle or Eilenberg telescope (see telescoping sum).

The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0:

1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 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.