Notice
Swindle? *****
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YES, you can swindle 
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 )
...
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 ).
= + ( 
+ ) + ( + ) + ... =
= + + + + ... = + + + + ... =
= + ( + ) + ( + ) + ... =
