9780023606922, Chapter 2 1, Problem 62E

Prévia do material em texto

Chapter 2.1, Problem 62E Step-by-step solution Step 1 of 2 Least upper bound of a set is unique To prove that the least upper bound of a set is unique, consider the definition of upper bound. A number a is said to be an upper bound of X a for The number a is said to be the least upper bound of X the numberb is also an upper bound of X, and Step 2 of 2 Consider a set X with two upper bounds, a and Let b be an upper bound and a be a least upper bound. Using the definition, Now, let a be an upper bound and b be the least upper bound. Therefore, by definition, b≤a. Hence, from the two inequalities, we geta=b. Hence, the two upper bounds are identical, or the least upper bound is unique. Therefore, the least upper bound of a set is unique.