Proving Pointwise Convergence for Sequence of Functions fn(x)

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In summary, the conversation discusses the concept of pointwise convergence and the function fn(x), which is defined differently depending on the value of x. The speaker expresses difficulty in proving that fn(x) converges to the zero function, f(x)=0. They suggest starting by choosing specific values for x0 and epsilon and finding a value for N that will satisfy the convergence condition.
  • #1
doggie_Walkes
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This has stuck with me for a long time, i just can't do it.

If the sequence of functions, fn(x): R+ --> R

where fn(x) is defined as

fn(x) = x/n if 0 is greater than and equal to x, x is less than and equal to n

fn(x) = 1 if x is strickly greater than n


I need to show that the fn(x) is pointwise convergence so the zero function. f(x) =0
 
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  • #2
Pick any x0>0 and epsilon>0. Find an N such that for all n>N, f_n(x0)<epsilon. Surely you can at least TRY and start.
 

FAQ: Proving Pointwise Convergence for Sequence of Functions fn(x)

What is pointwise convergence?

Pointwise convergence is a mathematical concept that describes the behavior of a sequence of functions. It means that as the independent variable approaches a certain value, the values of the functions in the sequence also approach a specific value.

How is pointwise convergence different from uniform convergence?

Pointwise convergence and uniform convergence are both ways to describe the behavior of a sequence of functions. The main difference is that pointwise convergence only requires the functions to approach a specific value at each point, while uniform convergence requires the functions to approach the same value at every point in the domain.

What is the importance of pointwise convergence in mathematical analysis?

Pointwise convergence is a fundamental concept in mathematical analysis that allows us to study the behavior of sequences of functions. It is used to prove important theorems, such as the Weierstrass M-test, which is essential in determining the convergence of infinite series.

How is pointwise convergence tested?

To test for pointwise convergence, we evaluate the function at each point in the domain and see if the sequence of values approaches a specific value as the independent variable approaches that point. If this is true for every point in the domain, then the sequence of functions is pointwise convergent.

What are some real-world applications of pointwise convergence?

Pointwise convergence has many applications in various fields, including physics, engineering, and economics. For example, it is used in analyzing wave functions in quantum mechanics, determining the stability of structures in engineering, and modeling the behavior of financial markets in economics.

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