I need help with this question,
Does any open interval in R have a maximum? Explain your answer.
Thomas
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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
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?
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.