Prove that elements in a neighborhood have their own neighborhood?

  • Thread starter Eclair_de_XII
  • Start date
  • Tags
    Elements
In summary, the proof is saying that if y is an element of a neighborhood of x, then there exists a neighborhood of y such that (y-\delta,y+\delta) is a subset of (x-\epsilon,x+\epsilon).
  • #1
Eclair_de_XII
1,083
91
Sorry for the abbreviation above; character limits, and all. Anyway:

1. Homework Statement

"Suppose that ##x∈ℝ## and ##\epsilon>0##. Prove that ##(x-\epsilon,x+\epsilon)## is a neighborhood of each of its members. In other words, if ##y∈(x-\epsilon,x+\epsilon)##, then prove that there is a ##\delta>0## such that ##y∈(y-\delta,y+\delta)⊆(x-\epsilon,x+\epsilon)##."

Homework Equations


Neighborhood: "A set ##O⊂ℝ## is a neighborhood of ##x∈ℝ## if ##O## contains an interval of positive length centered at ##x##. In other words, there exists an ##\epsilon>0## such that ##(x-\epsilon,x+\epsilon)⊆O##."

The Attempt at a Solution


Okay, so I get the feeling that I was supposed to do something else for this proof; specifically something I learned from this class. Here is the proof I wrote, otherwise:

"We start with the fact that ##x-\epsilon<y<x+\epsilon##. Subtracting ##x## from the inequality gives ##-\epsilon<y-x<\epsilon##, or ##|y-x|<\epsilon##. Then subtracting ##|y-x|## gives ##0<\epsilon-|y-x|##. Now we set ##0<\delta≤\epsilon-|y-x|##.

Now, we show that the interval ##(y-\delta,y+\delta)## is a non-empty subset of ##(x-\epsilon,x+\epsilon)##. We start with the inequality ##0<\delta≤\epsilon-|y-x|##.
- Adding ##y## gives ##y<y+\delta≤\epsilon-|y-x|+y##.
- Reversing the signs of the inequality and adding ##y## gives: ##y>y-\delta≥y-(\epsilon-|y-x|)##.

We take ##y-x≥0##. The case for if ##x-y≥0## is similar.

- Then the first inequality becomes: ##y<y+\delta≤\epsilon-|y-x|+y=\epsilon-(y-x)+y=\epsilon-y+x+y=x+\epsilon##.
- And then the second inequality becomes ##y>y-\delta≥y-(\epsilon-|y-x|)=y-(\epsilon-(y-x))=y-(\epsilon-y+x)=2y-(x+\epsilon)=(2y-2x)+x-\epsilon=2(y-x)+(x-\epsilon)≥x-\epsilon##

And so combining these statements yields: ##x-\epsilon≤y-\delta<y<y+\delta≤x+\epsilon##. And so there exists a ##\delta>0## such that if ##y## is an element of a neighborhood of ##x##, then there exists a neighborhood of ##y## such that ##(y-\delta,y+\delta)⊆(x-\epsilon,x+\epsilon)##."

I have a strong feeling that I was supposed to do something else, but cannot think of what it could be.
 
Last edited:
Physics news on Phys.org
  • #2
(I extended the title)

That works. You can do it more compact with the triangle inequality but analyzing the separate cases works as well.
Eclair_de_XII said:
Now we set ##0<\delta≤\epsilon-|y-x|##
I would write that as "now we choose a ##\delta## such that..." to make clear what you set. Alternatively you can explicitly define one (e.g. the average between the two sides).
 
  • #3
Gotcha. Thanks.
 

What is a neighborhood in the context of scientific research?

A neighborhood in scientific research refers to a group of elements or variables that are closely related or have similar properties. These elements are typically studied together in order to understand their interactions and behaviors.

Why is it important to prove that elements in a neighborhood have their own neighborhood?

Proving that elements in a neighborhood have their own neighborhood is important because it helps us better understand the complexity and interconnectedness of the natural world. It allows scientists to more accurately study and predict the behavior of these elements and their impact on larger systems.

How is it determined that elements in a neighborhood have their own neighborhood?

The determination that elements in a neighborhood have their own neighborhood is based on extensive research and analysis. Scientists may use various methods such as statistical analysis, experimentation, and observation to gather evidence and support their findings.

What are the potential implications of not recognizing that elements in a neighborhood have their own neighborhood?

If scientists fail to recognize that elements in a neighborhood have their own neighborhood, it could lead to inaccurate or incomplete conclusions about the behavior of these elements. This could have negative consequences for our understanding of natural systems and could hinder our ability to effectively address complex issues.

Can the concept of neighborhoods be applied to other fields besides scientific research?

Yes, the concept of neighborhoods can be applied to various fields such as sociology, economics, and urban planning. In these fields, neighborhoods refer to a group of people or entities that share common characteristics or are closely interconnected. The concept of neighborhoods can also be useful in understanding and addressing issues in these fields.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
536
  • Calculus and Beyond Homework Help
Replies
10
Views
316
  • Calculus and Beyond Homework Help
Replies
5
Views
864
  • Calculus and Beyond Homework Help
Replies
19
Views
1K
Replies
1
Views
615
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
Back
Top