1. The problem statement, all variables and given/known data I am working through an introductory real analysis textbook and am having a little trouble with certain aspects of the proof of the well ordering property (I am new to proving). Theorem: Every nonempty subset of the natural numbers (N) has a smallest element. Proof: Let S[itex]\subset[/itex]N and S≠∅. Then inf(S) must exist and be a real number because N has lower bound. If inf(S) [itex]\in[/itex] S, we are done. If inf(S) [itex]\notin[/itex] S, then it is the greatest lower bound of S but is not min(S). There must be an element x [itex]\in[/itex] S that is smaller than inf(S)+1 since inf(S) is the greatest lower bound of S. Thus we have inf(S) < x < inf(S)+1. -This is the first part I had trouble with. What is the justification for the fact that inf(S)+1 is greater than some x[itex]\in[/itex]S? From which axiom(s)/definition(s) can we infer that inf(S) is a real number that is greater than the greatest integer not in S? Since x > inf(S), it is not a lower bound of S. This means there must be another element y[itex]\in[/itex]S such that inf(S) < y < x < inf(S)+1. -Perhaps due to the earlier misunderstanding,the existence of such a y does not seem readily apparent to me. These two points are my main stumbling blocks and the rest of the proof is clear to me (contingent upon the existence of y). Any elucidation would be greatly appreciated.