1. The problem statement, all variables and given/known data Let X be a set and let f be a one-to-one mapping of X into itself such that [itex] f[X] \subset X [/itex] Then X is infinite. 3. The attempt at a solution Lets assume for the sake of contradiction that X is finite and there is an f such that it maps all of the elements of X to a proper subset of X called Y. And this function also preserves the 1-to-1 mapping. If Y is a proper subset then |Y|<|X| and if X is finite then Y is also finite, but there is a problem with the one to one mapping, if Y has less elements then X then there is no 1-to-1 mapping. So this is a contradiction of our original assumption therefore X is infinite.