Proving Openness of {f(x)>a} in R

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In summary, proving openness of {f(x)>a} in R means demonstrating that the set of all real numbers x for which f(x) is greater than a is an open set in the real number system. Openness of {f(x)>a} can be determined by using the definition of an open set and is important in understanding the behavior of the function f(x) and its relationship to the value a. The necessary steps to prove openness include defining the function, choosing a value for a, using the definition of an open set, and providing a logical argument. However, it is not possible to prove openness for all functions and values of a in R, as some functions may have properties that prevent the set from being open. Care
  • #1
Treadstone 71
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"Let (f_n) be an increasing sequence of continuous functions on R. Suppose [tex]\forall x\in\mathbb{R}(f(x)=\lim_{n\rightarrow\infty}f_n(x))[/tex], and suppose that [tex]f(x)<\infty[/tex] for all x, prove that [tex]\{x\in\mathbb{R}:f(x)>a\}[/tex] is open for all a in R."

I think an additional condition of uniform convergence is required.
 
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  • #2
No, you don't need uniform convergence. The proof is essentially a one- or two-liner.
 
  • #3
I got it. Thanks.
 

FAQ: Proving Openness of {f(x)>a} in R

What does it mean to prove openness of {f(x)>a} in R?

Proving openness of {f(x)>a} in R means demonstrating that the set of all real numbers x for which f(x) is greater than a is an open set in the real number system.

How is openness of {f(x)>a} determined?

Openness of {f(x)>a} can be determined by using the definition of an open set, which states that a set is open if every point in the set has a neighborhood contained within the set.

What is the importance of proving openness of {f(x)>a} in R?

Proving openness of {f(x)>a} in R is important because it allows us to make conclusions about the behavior of the function f(x) and its relationship to the value a. It also helps in understanding the properties of open sets in the real number system.

What are the necessary steps to prove openness of {f(x)>a} in R?

The necessary steps to prove openness of {f(x)>a} in R include defining the function f(x), choosing a value for a, using the definition of an open set to determine if every point in the set {f(x)>a} has a neighborhood contained within the set, and providing a logical argument to show that the set is indeed open.

Can openness of {f(x)>a} be proven for all functions and values of a in R?

No, it is not possible to prove openness of {f(x)>a} for all functions and values of a in R. Some functions may have discontinuities or other properties that prevent the set {f(x)>a} from being open. It is necessary to carefully analyze the function and value of a in order to determine if the set is open.

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