1. ## real numbers

I need help with this question,

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

Thomas

2. ## Re: 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?

3. ## Re: 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

4. ## Re: real numbers

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

So can $(a,b)$ have a maximal element?

5. ## Re: real numbers

Originally Posted by thomasboateng
I need help with this question,

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

Thomas

$J=(a,b)$

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

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

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

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

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

6. ## Re: 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 $N= \frac{M+b}{2}$.

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.