supremum

**inthequestofproofs** · February 28th 2010, 08:32 PM

Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?

**Drexel28** · February 28th 2010, 08:38 PM

Originally Posted by inthequestofproofs

Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?

I assume, as your last posts indicated, that you are dealing with closed subsets of $\mathbb{R}$ . Then the answer is yes. Notice that $\overline{E}=\left\{x\in\mathbb{R}:d(x,E)=0\right\ }$ . But, if $E$ is closed then $\overline{E}=E$ . So, let me ask you: what is $d\left(E,\sup\text{ }E\right)$ ?

(if the supremum exists)

**inthequestofproofs** · February 28th 2010, 08:40 PM

It would be zero, and thus, it is in the closed set

**Drexel28** · February 28th 2010, 08:43 PM

Originally Posted by inthequestofproofs

It would be zero, and thus, it is in the closed set

Now prove it

**hjortur** · March 1st 2010, 02:58 AM

I wanted to ask what about the set

$M=[0,\inf[$ , which is a closed subset of $\mathbb{R}$ .

Then does the supremum of $M$ exist?

if so then $sup\{M\}=\inf$ but would you say that it is in $M$ ?

What about the empty set? it is both open and closed ?

what is $sup\{\varnothing\}$ ? $- \inf$ ?

**HallsofIvy** · March 1st 2010, 03:05 AM

In order to have a supremum, a set must first have an upper bound. $[0, \infty[$ does not have an upper bound so no supremum.

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