A in R (real numbers) is a nonempty set. Show that A is bounded iff there exists a closed bounded interval such that [c,d] is in A. What does this proof look like?
Follow Math Help Forum on Facebook and Google+
Originally Posted by OntarioStud A in R (real numbers) is a nonempty set. Show that A is bounded iff there exists a closed bounded interval such that [c,d] is in A. What does this proof look like? Are you sure of the problem statement? Consider A = R. The closed bounded interval [0,1] is in A but A is not bounded. -Dan
OK. I realized my mistake. It should have read "such that A is in [c,d]." Obviously here, c would be a lower bound and d would be an upper bound, but how would you prove this?
Originally Posted by OntarioStud "such that A is in [c,d]." Obviously here, c would be a lower bound and d would be an upper bound, but how would you prove this? There really is nothing to prove rigorously. Given that A is a subset of [c,d], if x belongs to A then c<x<d. Therefore A is bounded. If A is bounded then let c=glb(A) and d=lub(A). By definition for all x in A, c<x<d. So that if follows that A is a subset of [c,d].
View Tag Cloud