Can You Solve a Problem Using the Definition of Supremum?

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Homework Statement
I have solved the problem below; however, I am unsure whether it is correct (I list the step below which I am not sure whether I am allowed to do).
Relevant Equations
##M = sup S##
##s > M - \epsilon##
For this problem,
1710202583981.png

My solution:
Using definition of Supremum,
(a) ##M ≥ s## for all s
(b) ## K ≥ s## for all s implying ##K ≥ M##

##M ≥ s##
##M + \epsilon ≥ s + \epsilon##
##K ≥ s + \epsilon## (Defintion of upper bound)
##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum
##M ≥ s + \epsilon##

Now we have two cases for the inequality:

(1) ##M = s + \epsilon##
(2) ##M > s + \epsilon##

For case (1), ##M ≠ s + \epsilon## from definition of Supremum so,

##M > s + \epsilon##
##M - \epsilon > s##

I am unsure why I get the wrong inequality direction. However, apart for that, it seems correct. I would appreciate any help.

Thanks for any help - Chiral.
 

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I wonder if you could use a proof by contradiction., i.e.,

Assume there exists an ##\epsilon>0## such that there does not exist a ##s\in S## such that##s>M-\epsilon##, then it must be that ##M \neq \text{sup}(S)##.
 
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ChiralSuperfields said:
Homework Statement: I have solved the problem below; however, I am unsure whether it is correct (I list the step below which I am not sure whether I am allowed to do).
Relevant Equations: ##M = sup S##
##s > M - \epsilon##

For this problem,
View attachment 341649
My solution:
Using definition of Supremum,
(a) ##M ≥ s## for all s
(b) ## K ≥ s## for all s implying ##K ≥ M##

##M ≥ s##
Is this any particular s? Suppose it is thousands below M. What will that prove?
ChiralSuperfields said:
##M + \epsilon ≥ s + \epsilon##
Is this any particular ##\epsilon##? What if it is huge?
ChiralSuperfields said:
##K ≥ s + \epsilon## (Defintion of upper bound)
No. If s is close enough to M and ##\epsilon## is huge, this is not true.
I think this approach is doomed to fail. You need to rethink it.
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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