Prove x belongs to the set or is an accumulation point.

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In summary, x is an accumulation point of S if and only if each neighborhood of x contains a member of S different from x.
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Homework Statement


Let S be a nonempty set of real numbers bounded from above and let x=supS. Prove x either belongs to the set or is an accumulation point of S.


Homework Equations


x is an accumulation point of S iff each neighborhood of x contains a member of S different from x. That is, every neighborhood of x contains infinitely many points of S.


The Attempt at a Solution


So, do I need to prove that if x is not a member of S, then it is an accumulation point? I am a little confused about how to go about this.

So, there would obviously be two possibilities. Either x is a member of S, or it is not. If not, we need to prove x is an accumulation point. Wouldn't we need to know that S is infinite though? Is so, wouldn't it be much like the proof of the Bolzano-Weierstrass theorem?

Any help would be greatly appreciated! Thanks!
 
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  • #2
So, I might have got it...

I tried it by contradiction.
Assume x is not a member of S and assume it is not an accumulation point of S. If there is a neighborhood (x - epsilon, x + epsilon) containing x that does not have a point of S, then (x - epsilon) is an upper bound of S that's less than x. This contradicts x being the LEAST upper bound of S. Therefore, we have found a contradiction and x is indeed an accumulation point of S.

Just a thought. Let me know if it's right!
 
  • #3
jrsweet said:
So, I might have got it...

I tried it by contradiction.
Assume x is not a member of S and assume it is not an accumulation point of S. If there is a neighborhood (x - epsilon, x + epsilon) containing x that does not have a point of S, then (x - epsilon) is an upper bound of S that's less than x. This contradicts x being the LEAST upper bound of S. Therefore, we have found a contradiction and x is indeed an accumulation point of S.

Just a thought. Let me know if it's right!

That's it exactly.
 
  • #4
Thanks Dick! :)
 

1. How do you prove that x belongs to a set?

To prove that x belongs to a set, you need to show that x satisfies the defining characteristics or properties of the set. This can be done by using logical reasoning, mathematical operations, or other relevant methods.

2. What is an accumulation point?

An accumulation point is a point in a set that has the property that for any neighborhood of the point, there exists at least one element of the set other than the point itself. In simpler terms, an accumulation point is a point where a set accumulates or clusters together.

3. How do you prove that x is an accumulation point of a set?

To prove that x is an accumulation point of a set, you must show that for any neighborhood of x, there exists at least one element of the set within that neighborhood. This can be shown through mathematical proofs or by using the definition of accumulation points.

4. Can a set have more than one accumulation point?

Yes, a set can have multiple accumulation points. This often occurs when the set has a dense or infinite structure, such as the real numbers or irrational numbers. In these cases, there may be an infinite number of accumulation points within the set.

5. Are accumulation points the same as limit points?

Yes, accumulation points and limit points are the same concepts and can be used interchangeably. Both refer to points within a set that have the property of being approached by elements of the set.

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