The proof is correct but may not be appropriate for all audiences. To begin with, the statement that a subset of {1, ..., n} has size <= n is completely obvious to most people and needs a proof only in certain circumstances, e.g., when you are working with specific axioms or when you want to explain the proof to a computer using a proof assistant. The amount of details in the proof is also overwhelming for human audience, in particular, for the fact that and imply . Next time, if your statement is too obvious or if your proof has too many or too little details, why don't you explain your reasons in the beginning?