Exercises
[07K] Prerequisites:[07C],[07D],[07F].(Proposed on 2022-10-13)
Let \(A⊆ X\) be a non-empty set; let\(I\) the smallest interval that contains \(A\); this is defined as the intersection of all intervals that contain \(A\) (and the intersection is an interval, by [07F]). Let \(M_ A\) be the family of majorants of \(A\), \(M_ I\) of \(I\); show that \(M_ A=M_ I\). In particular \(A\) is bounded from above if and only \(I\) is bounded from above; if moreover \(A\) has supremum, then \(\sup A=\sup I\). (Similarly for the minorants and infimum).
Solution 1