Computer Science - Discrete Math Review

1) Consider a proof that 1 + 1 = 1. Start by assuming a = b. Multiply both sides by a to obtain a2 = ab, then subtract b2 from both sides to show a2 b2 = ab b2. Then factor each side, obtaining (a + b)(a b) = b(a b). Then divide each side by (a b), obtaining a + b = b. Finally let both unknowns be 1 and our assumption a = b becomes true: 1 = 1; but our conclusion is false, 1 + 1 = 1. What error has this proof made? Please explain thoroughly. Thanks