Proving Nondecreasingness of Fn When Converging to F

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In summary, nondecreasingness of a function means that the function's output does not decrease as the input increases. It is important to prove nondecreasingness when a function is converging to a certain value to ensure its orderly and predictable behavior. Nondecreasingness also ensures that the limit of a function exists and is the same from both sides of the input value. To prove nondecreasingness mathematically, one must show that the output value at the second point is greater than or equal to the output value at the first point. Exceptions to nondecreasingness include functions with vertical asymptotes or removable discontinuities, and points of inflection.
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bobbarker
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Homework Statement


Prove: If {Fn} converges to F on [a,b] and Fn is nondecreasing for each n[tex]\in[/tex] N, then F is nondecreasing.

Homework Equations


n/a

The Attempt at a Solution


First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove uniformly and then it holds for pointwise as well?

My work so far:

Suppose Fn converges to F and Fn is nondecreasing: i.e., for all n [tex]\in[/tex] N, x1 [tex]\leq[/tex] x2 [tex]\Rightarrow[/tex] Fn(x1) [tex]\leq[/tex] Fn(x2).

So [tex]\forall[/tex] [tex]\epsilon[/tex] > 0, [tex]\exists[/tex] N1 [tex]\in[/tex]N, such that [tex]\forall[/tex] n [tex]\geq[/tex] N1, sup{|Fn(x)-F(x)|: x [tex]\in[/tex] [a,b]} < [tex]\epsilon[/tex].

Equivalently, ||Fn(x) - F(x)||[tex]_{}[a,b][/tex] < [tex]\epsilon[/tex]

| ||Fn(x)||[tex]_{}[a,b][/tex] - ||F(x)||[tex]_{}[a,b][/tex] | < [tex]\epsilon[/tex] by the reverse triang. inequal

||F(x)||[tex]_{}[a,b][/tex] - [tex]\epsilon[/tex] < ||Fn(x)||[tex]_{}[a,b][/tex] < ||F(x)||[tex]_{}[a,b][/tex] + [tex]\epsilon[/tex]

I feel like I'm heading down a dead end. Any ideas?
 
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  • #2


bobbarker said:
First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove uniformly and then it holds for pointwise as well?

Since a sequence of functions that converges uniformly also converges pointwise, you actually want to prove it for the case of pointwise convergence (A weaker hypothesis gives a more general theorem). It will also be much more direct to prove this way since you will be comparing two points and taking limits at those points.
 
  • #3


Here's a hint. Suppose f(c)>f(d) with c<d. Pick epsilon equal to |f(c)-f(d)|/2. Now remember what pointwise convergence means.
 

1. How do you define nondecreasingness of a function?

Nondecreasingness of a function means that the function's output (or value) does not decrease as the input (or independent variable) increases. This can also be described as the function being "monotonic" or "increasing".

2. Why is it important to prove nondecreasingness when a function is converging to a certain value?

Proving nondecreasingness when a function is converging to a certain value is important because it ensures that the function is approaching that value in an orderly and predictable manner. This can provide confidence in the accuracy and validity of the function's behavior.

3. What is the relationship between nondecreasingness and the limit of a function?

The limit of a function refers to the value that a function approaches as the input approaches a certain value. Nondecreasingness ensures that the limit of a function exists and is the same from both the left and the right side of the input value. In other words, the function does not suddenly jump to a different value when approaching the limit.

4. How do you prove nondecreasingness of a function mathematically?

To prove nondecreasingness of a function mathematically, you must show that for any two points on the function's domain (or input values), the output value of the function at the second point is greater than or equal to the output value at the first point. This can be shown using algebraic manipulations or calculus techniques, such as taking the derivative of the function and showing that it is always positive.

5. Are there any exceptions to a function being nondecreasing when it is converging to a certain value?

Yes, there are some exceptions to a function being nondecreasing when it is converging to a certain value. These exceptions include functions with vertical asymptotes or removable discontinuities, where the function can suddenly jump to a different value. Additionally, some functions may have points of inflection where the function changes from increasing to decreasing or vice versa.

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