I just purchased Tom Apostol's infamous calculus book. In the section on infimum and supremum, he notes, but does not prove, that sets such as all real z that satisfy z>x have no smallest member. Perhaps this is completely self evident (and it is) but many of his own proofs and those in the exercises involve proving things that are self evident to our intuition. The entire point, I suppose, is to verify that the axioms we have agreed to build our system on can at least be used to show things that are "self evident" to indeed be true. I have spent some time thinking about how this would be proved from the other axioms and theorems given in the book, but I am fairly new to the rigorous method. The opinions of those on this forum would be helpful. I imagine that any such proof would involve showing that a smallest member in such a set would make for a simple contradiction in inequalities, but, again, this concept seems so self evident that it is difficult to rid oneself of the stumbling block that is intuition.