To prove that 0 = 1, start with this series:
S = 1 - 1 + 1 - 1 + 1 - 1 + ...We can show that this equals 0 like so:
S = 1 - 1 + 1 - 1 + 1 - 1 + ... Original seriesS = (1 - 1) + (1 - 1) + (1 - 1) + ... Group each 1 - 1S = 0 + 0 + 0 + ... 1 - 1 = 0, so each (1 - 1) equals 0.S = 0 Adding zero to itself gives zero, so infinite zeroes are 0.We can also show that this equals 1 like so:
S = 1 - 1 + 1 - 1 + 1 - 1 + ... Original seriesS = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... Group each -1 + 1S = 1 + 0 + 0 + 0 + ... -1 + 1 = 0, so each (-1 + 1) equals 0.S = 1 Adding zero to itself gives zero, so infinite zeroes are 0. Add that to 1 to get 0.We can now show that 0 = 1 like so:
S = S S equals itselfS = 0 Earlier proofS = 1 Earlier proof0 = 1 Substitution of S for 0 and S for 1.This proves that 0 = 1.
Now that we know that 0 = 1, we can prove that any two numbers (A and B) are equivalent like so:
0 = 1 Earlier proof0 = B - A Multiply both sides by (B - A)A = B Add A to both sidesThus, A = B.