I need help with this question,

Does any open interval in R have a maximum? Explain your answer.

Thomas

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- Jul 30th 2011, 06:29 AMthomasboatengreal numbers
I need help with this question,

Does any open interval in R have a maximum? Explain your answer.

Thomas - Jul 30th 2011, 06:34 AMTKHunnyRe: real numbers
If it does, is it open? What say you?

There must be a defintion of "Open Interval" sitting about somewhere. Why not have a good, close look at it? - Jul 30th 2011, 06:48 AMthomasboatengRe: real numbers
i know it doesnt have any end points because you can always get a little bit more larger for example 0.1 then 0.11

but i dont know to explain correctly - Jul 30th 2011, 07:05 AMPlatoRe: real numbers
That may or may not be correct. It depends on which endpoint 0.1 is.

However, this is the essential point:*between any two real numbers there is a third number.*

If $\displaystyle x\in (a,b)$ then $\displaystyle a<x<b$.

Therefore $\displaystyle \left( {\exists y \in (x,b)} \right)\left[ {x < y < b} \right]$.

So can $\displaystyle (a,b)$ have a maximal element? - Jul 30th 2011, 07:25 AMAlso sprach ZarathustraRe: real numbers
- Jul 31st 2011, 01:59 AMHallsofIvyRe: real numbers
Indirect proof: Suppose the open interval, (a, b),

**does**have a maximum, M. Since M is in (a, b), a< M< b. Let $\displaystyle N= \frac{M+b}{2}$.

Prove:

1) N< b.

2) a< M< N

so N is in (a, b)

3) M< N, contradicting the hypothesis.

Do you understand the difference between a "maximum" and a "supremum" (least upper bound)? That is crucial here.