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

by splash_lover
Tags: alpha, alphasups, belongs, closure, equivalent, prove
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splash_lover
#1
Oct12-10, 01:35 PM
P: 11
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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radou
#2
Oct12-10, 02:01 PM
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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?
radou
#3
Oct12-10, 02:02 PM
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Of course, your "steps" are a correct way to prove equivalence of statements, from a logical point of view.

splash_lover
#4
Oct12-10, 06:03 PM
P: 11
Prove alpha=sup(S) is equivalent to alpha belongs to S closure

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.
radou
#5
Oct13-10, 11:42 AM
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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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