Let E be compact and nonempty. Prove that E is bounded and that sup E and inf E both belong to E.
For the first part can we use the fact that if E is compact iff it is closed and bounded? So since its compact than its closed and bounded..or is do I have to actually prove that its bounded? And also for the sup and inf, i'm not sure but would we set E=the interval and show that they are the inf and sup?
Hey this is what i've been working on. Can you let me know if i'm on the right track?
If E is compact then E is closed and bounded. Since E is nonempty, supE and infE both exist. To show supE∈ E, suppose not. Then supE ∈ Ec(compliment) which is an open set. Hence there exists δ > 0 such that
(supE − δ, supE + δ) ⊆ Ec(compliment). This is a contradiction because there must exist an element of E greater than
supE − δ....Then I just have to show it for inf..
That said, these ideas in most analysis courses are not pursued and so the Heine-Borel theorem (in some of my books) is an iff statement.