Exploring the Non-Existence of a Maximum in the Set (0,2)

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Homework Statement
I am trying to find the maximum of the set of real numbers in the open interval ##(0,2) ##
Relevant Equations
##(0,2)##
For this,
1700712007090.png

I am trying to understand why the set ##(0,2)## has no maximum. Is it because if we say for example claim that ##a_0 = 1.9999999999## is the max of the set, then we could come along and say that ##a_0 = 1.9999999999999999999999999999## is the max, can we continue doing that a infinite number of times so long that ##a_0 < 2##.

Many thanks!
 
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ChiralSuperfields said:
I am trying to understand why the set ##(0,2)## has no maximum. Is it because if we say for example claim that ##a_0 = 1.9999999999## is the max of the set, then we could come along and say that ##a_0 = 1.9999999999999999999999999999## is the max, can we continue doing that a infinite number of times so long that ##a_0 < 2##.
Yes. Within (0,2), you can get as close as you want to 2, but 2 itself is not in the set (0,2). The definition specifies that the maximum must be in the set. So (0,2) has no maximum point.
 
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I didn't understand your sentence after "max,".

Assume that ##(0,2)## has a maximum ##m<2.## Then ##m<m+\dfrac{2-m}{2}=\dfrac{2+m}{2}<2## which cannot be since ##m## was already the maximum. This is a contradiction, so there is no maximal number.

If you prefer the positive reasoning, then given any number ##m_0\in (0,2)## then ##m_1=\dfrac{2+m_0}{2}## is a number greater than ##m_0## and still smaller than ##2.## Now, we can proceed with that new number and define ##m_2= \dfrac{2+m_1}{2}.## This results in an infinite sequence
$$
0<m_0<m_1<m_2<\ldots < 2
$$
that gets closer and closer to ##2## but never ends. If this was what you wanted to say, then the answer is 'yes'.
 
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FAQ: Exploring the Non-Existence of a Maximum in the Set (0,2)

What does it mean for a set to have no maximum?

A set having no maximum means that there is no single element in the set that is greater than or equal to every other element. In other words, for any element you pick in the set, you can always find another element that is larger.

Why doesn't the set (0,2) have a maximum?

The set (0,2) represents all real numbers between 0 and 2, not including 0 and 2 themselves. Since there is no largest number that is less than 2 but greater than all other numbers in the set, the set does not have a maximum. For any number you choose within (0,2), you can always find another number closer to 2 but still less than 2.

Is there a difference between a maximum and a supremum?

Yes, there is a difference. A maximum is an actual element within the set that is greater than or equal to all other elements. A supremum (or least upper bound) is the smallest value that is greater than or equal to every element in the set but does not necessarily have to be an element of the set. For the set (0,2), the supremum is 2, but 2 is not included in the set, so it does not have a maximum.

Can a set have a supremum but no maximum?

Yes, a set can have a supremum but no maximum. The set (0,2) is a perfect example. The supremum of this set is 2, meaning 2 is the smallest number that is greater than or equal to every number in the set. However, since 2 is not an element of the set, the set does not have a maximum.

How does the concept of a maximum relate to open and closed intervals?

In an open interval like (0,2), the endpoints are not included, so there is no maximum. In a closed interval like [0,2], the endpoints are included, so the maximum would be 2. The inclusion or exclusion of endpoints in an interval determines whether or not the set can have a maximum.

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