Proving the Existence of Frontier Points for Sets in ℝ

In summary, the statements are being made: (i) If a and b are both in E and a is less than b, then the intersection of the closed interval [a, b] and the frontier of E is not empty. (ii) If a and b are both in E and a is less than b, then the intersection of the open interval (a, b) and the frontier of E is not empty. To prove or disprove these statements, it is necessary to show that there exists at least one point in the interval that is also in the frontier, or to provide a counterexample. This can be done without a drawing by using the definition of frontier as the set of all points where sequences in E and
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3.4. Let E[itex]\in[/itex]  R. Prove or disprove the following statements:
(i) if a[itex]\in[/itex]E and b[itex]\in[/itex]E[itex]^{c}[/itex] = ℝ\E and a < b then [a ,b] [itex]\cap[/itex]∂E IS NOT EQUAL TO ∅.
(ii) if a[itex]\in[/itex]E and b[itex]\in[/itex]E[itex]^{c}[/itex] = ℝ\E and a < b then (a ,b) [itex]\cap[/itex]∂E IS NOT EQUAL TO ∅.


I am really stuck I know that the frontier of a set is when a sequence in E and a sequence in E[itex]^{c}[/itex] converge to the same limit.
 
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  • #2
I am able to show this from drawing a sketch but how can I do it without a drawing?
 

FAQ: Proving the Existence of Frontier Points for Sets in ℝ

What is a "Frontier points proof"?

A "Frontier points proof" is a scientific method used to analyze and prove the validity of a theoretical concept or hypothesis. It involves identifying key points or pieces of evidence that support the concept and presenting them in a logical and persuasive manner.

How is a "Frontier points proof" different from other scientific proofs?

Unlike other scientific proofs that may rely on experimental data or statistical analysis, a "Frontier points proof" focuses on identifying and analyzing key points or evidence that support a concept or hypothesis. It is particularly useful in fields where experimental data may be limited or difficult to obtain.

What are the main steps involved in conducting a "Frontier points proof"?

The main steps in conducting a "Frontier points proof" include identifying the key points or evidence that support the concept, organizing them in a logical and persuasive manner, and analyzing the strength of each point. The proof should also address any potential weaknesses or counterarguments to the concept.

What types of concepts or hypotheses can be proven using a "Frontier points proof"?

A "Frontier points proof" can be used to prove a wide range of concepts or hypotheses, including those in the fields of physics, biology, psychology, and economics. It is particularly useful for proving theoretical concepts or ideas that may be difficult to test through traditional experimental methods.

What are the benefits of using a "Frontier points proof" in scientific research?

One of the main benefits of using a "Frontier points proof" in scientific research is that it allows researchers to prove theoretical concepts or hypotheses that may be difficult to test through traditional methods. It also encourages critical thinking and analysis of key points and evidence, leading to a stronger and more convincing argument for the concept being proved.

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