Pointwise Limits and Uniform Convergence Help

In summary, the conversation discusses the pointwise limits and uniform convergence of the sequences (fn), (1/n fn), and (fn2). The individual discussing the topic has determined that all three sequences are uniformly convergent, but is unsure if their calculations are correct. They provide their limits and working, but mention they are struggling to use latex. Another individual confirms the correctness of the limits and provides proofs for the first and third sequences, stating that the second one is not uniformly convergent. They also offer advice to keep thinking about uniform convergence until it becomes clear.
  • #1
BlackTiger
2
0
Hi all
Came across this question which i have attempted to answer but am sure i have gone wrong somewhere, any help would be appreciated...

Suppose fn(x):R-->R, where fn(x)=x+(1/n) (n belongs to N). Find the pointwise limits of the sequences (fn), (1/n fn) and (fn2). In each case determine whether the convergence is uniform.

Now i managed to get all three as being uniformly convergent and am sure i went wrong somewhere. I am still struggling to understand uniform convergence completely. The pointwise limits i computed were x, 0, x2 respectively...sorry i havnt put my working up struggling to use latex.
 
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  • #2
Your limits are right. (Let [tex]f[/tex] be the limit of [tex]f_n[/tex]). The first sequence is uniformly convergent: given [tex]\epsilon >0[/tex], choose N to be the lowest integer that is greater than:[tex]\frac{1}{\epsilon}[/tex].

Then, for all [tex]n>N[/tex]:
[tex]|f-f_n|=|x-(x+\frac{1}{n})|=\frac{1}{n}<\frac{1}{N}<\epsilon[/tex].

For the second one, note that [tex]sup_x|f_n|[/tex] does not go to 0. This tells you it is not uniformly convergent. To prove this, suppose we are given [tex]\epsilon >0[/tex]. For any choice of N, let [tex]x>\epsilon .N-\frac{1}{N}[/tex].
Then:

[tex]|f-f_N|=|0-\frac{x}{N}-\frac{1}{N^2}|=|\frac{x}{N}+\frac{1}{N^2}|>\frac{\epsilon .N-\frac{1}{N}}{N}+\frac{1}{N^2}=\epsilon[/tex].

The third one is very similar. I'm afraid all this might be a bit bewildering if you don't yet understand uniform convergence though! Keep thinking about it and eventually it will click.

PS - I may have made some mistakes here as I did it very quickly, so be sure to check everything!
 
Last edited:

Related to Pointwise Limits and Uniform Convergence Help

1. What is a pointwise limit?

A pointwise limit is the value that a function approaches at a specific point as the independent variable approaches that point. It is the output of a function at a specific input value.

2. How is a pointwise limit different from a limit?

A pointwise limit is specific to a particular point, while a limit is a general concept that describes the behavior of a function as the input approaches a certain value. In other words, a pointwise limit is a single value, while a limit can have different values at different points.

3. What is the definition of uniform convergence?

Uniform convergence is a property of a sequence of functions where the functions approach the same limit at the same rate, as the independent variable increases without bound. This means that the functions do not deviate significantly from each other as the input increases.

4. How is uniform convergence different from pointwise convergence?

Uniform convergence is a stronger type of convergence than pointwise convergence. In pointwise convergence, the functions may vary significantly from each other at different points. In uniform convergence, the functions stay close to each other over the entire range of the independent variable.

5. Why is uniform convergence important?

Uniform convergence is important because it guarantees that the limit of a sequence of functions is a continuous function. This is essential in many areas of mathematics and physics, where continuity is a crucial property. It also allows for easier analysis and manipulation of functions, as they behave in a more predictable manner.

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