1. ## Math Analysis

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?

2. 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

3. 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?

4. 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].