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Proof that 0.111... = 0

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Firstly,

 

as 9 + 1 = 10 is trivially true. Then, as

 

(see Proof that 0.999... equals 1) we can equate both sides leaving

 

which, after subtracting 0.999... from both sides cancels to:

 


QED. *cough*

Update: See here.

Solution to P=NP

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QED.