Note: Problems 1 - 3 are stated for subsets S of a general ordered field (F,

Problem 1. If there exists a: E S such that y S a: for every y E S, we say that :1: is the largest element of S, or write .11 = max S. (i) Prove that if S has a largest element then it is unique. (i) Prove that if S has a largest element then S has a supremum and sup S = maxS. (iii) Prove that if S is finite then S has a largest element. (iv) Give two examples of a set S which does NOT have a largest element: one which has a supremum (i.e. supS exists in ]F') and one which does not. Be sure to specify the eld IF