# Uniform Convergence: Does Not Converge on (0,1)

• hth
In summary, the conversation discusses a function, fn(x) = 1/(nx+1), on the interval (0,1) and how it does not converge uniformly. The reason for this is that although fn(x) approaches zero monotonically, it is not continuous on a compact interval. The conversation also mentions using an epsilon value and choosing an arbitrary x to show this formally.
hth

## Homework Statement

Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly.

## The Attempt at a Solution

I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not continuous on a compact interval. How would go about showing this formally/by example?

A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?

snipez90 said:
A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?

Isn't that the solution to it if f_n(x) = x^(n)? How does that apply here?

## What is uniform convergence?

Uniform convergence is a type of convergence in which a sequence of functions converges to a single function on a given interval. It means that the distance between the function and the limiting function can be made arbitrarily small for all points in the interval by choosing a large enough term in the sequence.

## How is uniform convergence different from pointwise convergence?

Pointwise convergence means that for every point in the interval, the sequence of functions converges to the limiting function. However, in uniform convergence, the convergence is not only at individual points, but also across the entire interval. This means that the convergence is uniform or "consistent" throughout the interval.

## Can a sequence of functions converge uniformly on an open interval?

Yes, a sequence of functions can converge uniformly on an open interval. This is known as uniform convergence on an open interval. It is important to note that the limiting function must be defined and continuous on the entire open interval for this convergence to occur.

## What does it mean for a sequence of functions to not converge uniformly on an interval?

If a sequence of functions does not converge uniformly on an interval, it means that there exists a point in the interval where the distance between the function and the limiting function cannot be made arbitrarily small, regardless of how large the term in the sequence is chosen. In other words, the convergence is not consistent throughout the interval.

## Can uniform convergence be used to prove continuity of a function?

Yes, uniform convergence can be used to prove continuity of a function. If a sequence of functions converges uniformly to a function on a given interval, and each function in the sequence is continuous on that interval, then the limiting function will also be continuous on that interval. This is known as the uniform limit theorem.

• Calculus and Beyond Homework Help
Replies
26
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
635
• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Topology and Analysis
Replies
2
Views
725
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
1K