Completeness of the Real Numbers

Definition 1) A nonempty set S of real numbers is said to be bounded above (below) if there exists some real number k s.t x≤k (x≥k) for all x in S. k is called a upper bound (lower bound) for S. Archimedes' Principle) Let ε, M be any two positive real numbers, There exists a k in N s.t M
A supremum (infimum) of S, denoted sup S (inf S) is a real number k s.t x≤k for all x in S, and k≤M if M is an upper bound ...