2. For any n E N, let G, denote the interval (-). Prove that each G, is an open set in R and each G, is not a closed set in R. Let G =N Gm. Lastly, explain why G is not an open set. (You are showing that an infinite intersection of open sets is not necessarily open.) Hint: For latter question, find all the elements in G.
2. For any n E N, let G, denote the interval (-). Prove that each G, is an open set in R and each G, is not a closed set in R. Let G =N Gm. Lastly, explain why G is not an open set. (You are showing that an infinite intersection of open sets is not necessarily open.) Hint: For latter question, find all the elements in G.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. For any \( n \in \mathbb{N} \), let \( G_n \) denote the interval \( \left(-\frac{1}{n}, \frac{1}{n}\right) \). Prove that each \( G_n \) is an open set in \( \mathbb{R} \) and each \( G_n \) is not a closed set in \( \mathbb{R} \). Let
\[
G = \bigcap_{n=1}^{\infty} G_n.
\]
Lastly, explain why \( G \) is not an open set. (You are showing that an infinite intersection of open sets is not necessarily open.)
*Hint: For latter question, find all the elements in \( G \).*
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This exercise explores the properties of open and closed sets in the context of real analysis, specifically focusing on the behavior of infinite intersections of open intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F805c2cc2-6060-43ca-959b-a3334a4a8475%2Fb85c67f6-a064-48b6-8fc5-ddb208bad492%2Flu6u1ay_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. For any \( n \in \mathbb{N} \), let \( G_n \) denote the interval \( \left(-\frac{1}{n}, \frac{1}{n}\right) \). Prove that each \( G_n \) is an open set in \( \mathbb{R} \) and each \( G_n \) is not a closed set in \( \mathbb{R} \). Let
\[
G = \bigcap_{n=1}^{\infty} G_n.
\]
Lastly, explain why \( G \) is not an open set. (You are showing that an infinite intersection of open sets is not necessarily open.)
*Hint: For latter question, find all the elements in \( G \).*
---
This exercise explores the properties of open and closed sets in the context of real analysis, specifically focusing on the behavior of infinite intersections of open intervals.
Expert Solution
Step 1
Here =
so,
=
Since all are Open intervals , so each is open set in R. because every point is interior point.
Now, a set is said to be closed iff it contains all of its limit points.
set of limit point for each =
since does not contain its limit set.so, is not closed.
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