This was posted on the old board. I believe there wasn't any satisfying conclusion to this "proof". This is also to meaninglessly increase my post count and copy & paste a copyrighted piece, I think, and forgetting to name the source.
Proof that 0 equals 1
The following is a "proof" that 0 equals 1:
0 = 0 + 0 + 0 + ...
= (1 − 1) + (1 − 1) + (1 − 1) + ...
= 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... (associative law)
= 1 + 0 + 0 + 0 + ...
= 1
Q.E.D.
The error here is that the associative law cannot be applied freely to infinite sums unless they are absolutely convergent. In fact, it is possible to show that in any field, 0 is not equal to 1.
Proof that 0 equals 1
The following is a "proof" that 0 equals 1:
0 = 0 + 0 + 0 + ...
= (1 − 1) + (1 − 1) + (1 − 1) + ...
= 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... (associative law)
= 1 + 0 + 0 + 0 + ...
= 1
Q.E.D.
The error here is that the associative law cannot be applied freely to infinite sums unless they are absolutely convergent. In fact, it is possible to show that in any field, 0 is not equal to 1.