Why Does the Set (0,2) Have No Maximum?

[member 731016]

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,

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!

 

[FactChecker]

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.

 

[fresh_42]

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'.