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Possible Proof That 1 + 1 Does Not Equal 2

Date: 04/19/2001 at 01:57:44
From: Desmond Leung
Subject: A possible proof for 1+1 not equal 2

A buddy of mine told me how to prove that 1 + 1 does not equal 2, and 
some say he was correct. I just want verification.

It goes something like this:

     1 = 1/3 + 1/3 + 1/3
     1 = 0.333... + 0.333... + 0.333...     (... means repeated)
     1 = 0.999...


     lim    (1)^n = 1


     lim    (0.999...)^n = 0


     lim    [(0.999...)^n + (0.999...)^n] = 0

which does not equal 2.

Would this work?

Date: 04/19/2001 at 14:27:47
From: Doctor TWE
Subject: Re: A possible proof for 1+1 not equal 2

Hi and thanks for writing to Dr. Math.

Here's the problem with this "proof."

     lim   (0.999...9)^n = 0

is true only if there is a *finite* number of 9's in the value in 
parentheses. With an infinite number of 9's, the expression becomes:

     lim   [sum(9/10^k)]^n
     n->oo  k=1

and we cannot conclude that this expression goes to zero.

For more information on false "proofs" of this sort, see the Dr. Math 
FAQ at:   

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum   
Associated Topics:
High School About Math
High School Calculus
High School Sequences, Series

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