I'm continuing through Dirac's book, The Principles of Quantum Mechanics. You can view this as a google book in the link below. http://books.google.com.au/books?id=...page&q&f=false [Broken] On page 28-29 he proves this Theorem: If ξ is a real linear operator and ξm|P> = 0 (1) for a particular ket |P>, m being a positive integer, then ξ|P> = 0 ... And the proof is fine by me except for the first part where he says: put m = 2 then (1) gives <P|ξ2|P> = 0 (2) and goes on to prove it by induction. But I don't see how (2) is 'given' from (1). Can anyone explain? If he is relying on assuming (1) is true for some ket |P> and positive integer m, then I just pick m = 1 and it's not a theorem. And besides, in this case he never demonstrates a definite example of the statement being true, which you need to do in induction.