Provide an example of each of the following or explain why it is impossible (1)

ID: 3145055 • Letter: P

Question

Provide an example of each of the following or explain why it is impossible

(1) A bounded set of rational numbers whose l.u.b is not rational

(2) A bouded set of integers whose l.u.b is not an integer

(3) A set of real numbers X1, X2, X3,... satisying three properties

(i) each Xn is a closed interval

(ii) for each n, Xn contains Xn+1

(iii) the intersection from 1 to infinity of Xn does not equal the empty set

(4) A set of real numbers X1, X2, X3,... satisying three properties

(i) X1 = (0,10)

(ii) each Xn is an open interval

(iii) for each n, Xn contains Xn+1

(iv) the intersection from 1 to infinity of Xn does not equal the empty set

(1) The set {x:x2<2} does not have a least upper bound in .

(2) No such set exist because the supremum of a bounded set of integers is an integer in the set.

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.