Real Analysis Convergence Question

In summary, to solve the question, let ε = |b-a|/2 and assume x ∈ Vε (a) and Vε (b). By contradiction, it can be shown that |b-a| = |(b-x)+(x-a)| < |b-a|/2 + |b-a|/2 = |b-a|, thus proving that there exists a number ε > 0 for which the ε-neighborhoods Vε (a) and Vε (b) are disjoint.
  • #1
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show that if a and b are distinct real numbers, then there exists a number ε > 0 such that the ε -neighorboods Vε (a) and Vε (b) are disjoint.

How to solve this question?

Thank you
 
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  • #2
That's a pretty simple question. What have you thought about it so far?
 
  • #3
vanhees71 said:
That's a pretty simple question. What have you thought about it so far?



would it be the following?

let |b-a|/2 = ε
Assume x ∈ Vε (a) and Vε (b).
|b-a| = |(b-Xn)+(Xn-a)| ≦ |(b-Xn)| + |(Xn - a)| < |b-a|/2 + |b-2|/2 = |b - a|
A contradiction
 
  • #4
Your notation isn't great - You use an Xn when you previously defined x, and you have a |b-2| when you probably mean |b-a|, but other than that it looks good to me.
 

1. What is the definition of convergence in real analysis?

Convergence in real analysis refers to the behavior of a sequence of numbers or functions as the number of terms or inputs increases. A sequence is said to converge to a limit if the terms get closer and closer to a specific value as the sequence progresses.

2. How do you prove the convergence of a sequence in real analysis?

The convergence of a sequence in real analysis can be proven using various methods such as the limit comparison test, ratio test, root test, and direct comparison test. These tests involve evaluating the limit of the sequence and checking for certain conditions to determine if the sequence converges or diverges.

3. What is the difference between pointwise and uniform convergence in real analysis?

Pointwise convergence in real analysis refers to the behavior of a sequence of functions at each individual point in the domain. Uniform convergence, on the other hand, refers to the behavior of a sequence of functions over the entire domain. In other words, a sequence of functions is pointwise convergent if it converges at every point, while it is uniformly convergent if it converges to the same limit at every point.

4. Can a sequence of functions converge pointwise but not uniformly?

Yes, it is possible for a sequence of functions to converge pointwise but not uniformly. This can happen if the convergence of the sequence is dependent on the specific point in the domain, rather than the entire domain. In such cases, the convergence may not hold for all points in the domain, leading to non-uniform convergence.

5. Why is convergence an important concept in real analysis?

Convergence is a fundamental concept in real analysis as it allows us to study the behavior of sequences and series of numbers or functions. It is also crucial in determining the convergence of integrals and derivatives, which are essential tools in many branches of mathematics and science. Understanding convergence is necessary for developing rigorous mathematical proofs and analyzing the behavior of mathematical models.

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