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

by splash_lover
Tags: alpha, alphasups, belongs, closure, equivalent, prove
 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.....
 HW Helper P: 3,224 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?
 HW Helper P: 3,224 Of course, your "steps" are a correct way to prove equivalence of statements, from a logical point of view.
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.
 HW Helper P: 3,224 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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