Prove alpha=sup(S) is equivalent to alpha belongs to S closure

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Homework Help Overview

The discussion revolves around proving the equivalence of two conditions related to the supremum of a set S of real numbers: that alpha equals sup(S) and that alpha belongs to the closure of S. The subject area is real analysis, focusing on properties of upper bounds and closures in the context of sets of real numbers.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to prove the equivalence by establishing two implications. They seek assistance specifically with the second step, which involves assuming that alpha belongs to the closure of S. Participants discuss the logical structure of the proof and the properties of closure and supremum.

Discussion Status

Some participants have provided insights regarding the properties of points in the closure of a set and the significance of the supremum. There is an ongoing exploration of how to effectively use these properties to advance the proof, with no explicit consensus reached yet.

Contextual Notes

Participants are considering the implications of the definitions of closure and supremum, and there may be constraints related to the completeness of the original poster's understanding of these concepts.

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Given that alpha is an upper bound of a given set S of real numbers, prove that the following two conditions are equivalent:
a) We have alpha=sup(S)
b) We have alpha belongs to S closure

I'm trying to prove this using two steps.
Step one being: assume a is true, then prove b is true.
Step two being: assume b is true, then prove a is true.

Could anyone help me with step two?
Assuming alpha belongs to S closure...
 
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If I remember right, I think I gave you a useful condition for a point to be in the closure of a set. Do you see how you can use it here?
 
Of course, your "steps" are a correct way to prove equivalence of statements, from a logical point of view.
 
No I don't see how I can use it here in this problem.

How would I start my step two? I know I assume alpha belongs to S closure, but I am not sure where to go from there.
 
A point x is in the closure of a set A if any neighbourhood of x intersects A. Now, what is an important property of the supremum (involving "ε")?
 

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