I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets. Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A. Then by definition "For every x belongs to A, x is lesser than or equal to S" Let M be an upper bound such that M is less than S and M does not belong to A(Assertion 1) Then, it leads to contradiction because A is bounded above by M and M is less than least upper bound S, which means an upper bound is less than least upper bound. So, we have either M=S or M>S. So, Assertion 1 is false. Then by converse of Assertion 1, if N is less than S, it may belong to A but can't be an upper bound or if N is less than S, it may not belong to A. (Assertion 2) Is my logic right?