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Swindle? *****

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YES, you can swindle Wink and prove theorems similar to this:

 

Swindle can be correctly used in many proofs, e.g.

Irreducibility of the sphere

Cantor–Bernstein–Schröder theorem ( Wiki EN )

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Can you add the next one to the list?

A famous swindle (atributed to Samuel Eilenberg and Barry Mazur, Wiki EN) uses a foolish

1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0

identity idea to prove various correct results in a true correct way. Read great EXPO and a DISCUSSION on Terrence Tao blog ( EN ).

 

Smile = Smile + ( Sad + Smile ) + ( Sad + Smile ) + ... =

= Smile + Sad + Smile + Sad + ... = Sad + Smile + Sad + Smile + ... =

= Sad + ( Smile + Sad ) + ( Smile + Sad ) + ... = Sad