How do I prove that a sequence is open?

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In summary, the conversation discusses the concept of an open set, where for every point in the set, there exists a positive distance such that all points within that distance are also in the set. The conversation then goes on to discuss proving that a sequence within an open set is also open, by choosing a real number N such that all points in the sequence after N are within the chosen distance from the point x in the set. It is concluded that this, along with the openness of the set, proves the required result.
  • #1
lolalyle
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Homework Statement



Let B be an open set. Let x \in A. Let sn be a sequence such that lim sn=x. Then there exists an N such that sn \in A for all n>N.

Definition of Open Set: S is open if for every x \in S, there exists an E>0 such that (s-E,s+E) \subset S

Homework Equations



Prove that the sequence is open.


The Attempt at a Solution



Let E>0 be given.
Let x \in A be given.
Let lim sn=x.
Choose N such that ... for all n>N.
...
 
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  • #2
The first sentence of your post should be "Let A be an open set" I guess..
[tex] .\quad A [/tex]
(--------x-)

And then you have a sequence of number that converge to x. What is the definition of a sequence converging to a number?
 
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  • #3
For all E>0, there exists a real number N such that for all n in the natural numbers, n>N implies that abs(sn-s)<E.
 
  • #4
So basically you can get as close to x as you want as long as you chose a N big enough.

Do you see that this together with that A is open will give you the required result?
 
  • #5
Yes! Thank you. I definitely do not know why I didn't see that in the first place.
 
  • #6
If a sequence is not bounded does it have a maximum?
 

FAQ: How do I prove that a sequence is open?

1. What does it mean for a sequence to be open?

A sequence is considered open if every point in the sequence has a neighborhood that is also contained within the sequence.

2. How can I prove that a sequence is open?

To prove that a sequence is open, you can use the definition of an open set and show that for every point in the sequence, there exists a neighborhood of that point that is also contained within the sequence.

3. Can a sequence be both open and closed?

No, a sequence cannot be both open and closed. A sequence is considered open if every point has a neighborhood contained within the sequence, while a sequence is considered closed if it contains all of its accumulation points.

4. Why is it important to prove that a sequence is open?

Proving that a sequence is open is important because it allows us to determine if the sequence is a subset of a larger set, and if it has any accumulation points. This information can be used to analyze the behavior and properties of the sequence.

5. Are there any alternative methods for proving that a sequence is open?

Yes, there are alternative methods for proving that a sequence is open. One method is to use the concept of interior points, where a point is considered an interior point if it has a neighborhood contained within the sequence. Another method is to use the concept of boundary points, where a point is considered a boundary point if every neighborhood of the point contains points both in and outside of the sequence.

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