Proof that one equals two

A maths teacher showed me this a long ago. At the time it took me a while to figure out why it wasn't right (which clearly it isn't!)

Start with the following simple equation:

a = b

Multiply both sides by a

a² = ab

Subtract b² from both sides

a² - b² = ab - b²

Factorise

(a + b)(a - b) = b(a - b)

Divide both sides by (a - b)

a + b = b

Now, since at the start we said a = b

2b = b

Dividing both sides by b, gives:

2 = 1

Q.E.D.

Proof that One Equals Two