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?

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- September 25th 2006, 02:12 PMOntarioStudMath 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? - September 25th 2006, 02:16 PMtopsquark
- September 28th 2006, 08:18 AMOntarioStud
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?

- September 28th 2006, 09:08 AMPlato
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].