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    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?
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    Quote Originally Posted by OntarioStud View Post
    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
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    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?
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    Quote Originally Posted by OntarioStud View Post
    "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].
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