# How do I prove that a sequence is open?

• lolalyle
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.
lolalyle

## 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.
...

The first sentence of your post should be "Let A be an open set" I guess..
$$.\quad A$$
(--------x-)

And then you have a sequence of number that converge to x. What is the definition of a sequence converging to a number?

Last edited:
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.

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?

Yes! Thank you. I definitely do not know why I didn't see that in the first place.

If a sequence is not bounded does it have a maximum?

## 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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