this is a common "proof" that is often bandied about, even though it's wrong.
Let a=b.
multiply by a:
a^2 = ab
subtract b^2:
a^2 - b^2 = ab - b^2
factor both sides:
(a+b)(a-b) = (a-b)b
cancel (a-b)
(a+b) = b
substitute a=b from the first step:
2b = b
divide by b:
2=1
subtract 1 from both sides
1=0
fortunately sanity is saved when you realize that the step where you cancel (b-a) from both sides is actually dividing by zero, since you specify beforehand that b=a. this is not so much a proof as a mathematical fallacy. see the link below for another, more complicated version of the same thing.