Is the following true about sups and infs?

  • #1
I am trying to determine whether the following is true.

Let [itex]A\subset \mathbb R[/itex], and let [itex]|A| = \{|a|:a\in A\}[/itex]. Then [itex]\sup |A| = \max \{|\inf A|, |\sup A|\}[/itex].

I think I could figure this out on my own, but there is something else demanding my attention right now...so, hopefully this isn't too tough. Thanks!
 

Answers and Replies

  • #2
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Seems true... But it still needs to be proven. How would you begin such a proof?
 
  • #3
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I'm on the phone now, but I can't help but read this :p

I think, show different cases: if |inf A| < |sup A|, show that that sup|A| = |sup A|; etc.
 
Last edited:
  • #4
HallsofIvy
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First note that, for any a in A, [itex]inf(A)\le a\le sup(A)[/itex] so that [itex]|a|\le max(|sup(A)|, |inf(A)|)[/itex] so that "max(|sup(A)|, |min(A)|)" is an upper bound. Now, to show it is the least upper bound, show that there must be a in A such that |a| is arbitrarily close to either |sup(A)| or |inf(A)|, whichever is larger.
 

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