real numbers

I need help with this question,

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

Thomas

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?

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

Quote:

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Originally Posted by thomasboateng

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

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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?

Quote:

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Originally Posted by thomasboateng

I need help with this question,

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

Thomas

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$\displaystyle J=(a,b)$


$\displaystyle \text{inf}\{ J \}=a$

$\displaystyle \text{sup}\{ J \}=b$

$\displaystyle a,b \not\in (a,b)$, hence by definition:

$\displaystyle \text{min}\{ J \}\neq a$

$\displaystyle \text{max}\{ J \}\neq b$

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.