Proving the Union of Intervals is All of N

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Homework Help Overview

The problem involves proving that the union of intervals [1,n] from n=1 to n=infinity encompasses all natural numbers N. The discussion revolves around the definitions and properties of intervals and natural numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the use of induction and the Archimedean property, while questioning the validity of the original claim regarding the union of intervals. There is also a discussion about the definitions of the intervals and natural numbers.

Discussion Status

The discussion is ongoing, with some participants questioning the original statement and suggesting that the union may not contain only natural numbers. Clarifications about definitions are being sought, indicating a productive exploration of the topic.

Contextual Notes

There is a mention of a counterexample involving the number 3/2, which raises concerns about the completeness of the original claim. The definitions of the intervals and natural numbers are also under scrutiny.

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Homework Statement


Prove that the union of intervals [1,n] from n=1 to n=infinity is all of N.

The Attempt at a Solution



Do I use induction on this? Archimedes? (This question is before the section of Archimedes though). I need help on how to start it!
 
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Counterexample:
3/2 is in the union, because it is in [1, n] for, for example, n = 2. But 3/2 is not a natural number.

Did you mean: "prove that the union contains N"?
 
I mean that the union of all those intervals from 1 to infinity IS N.
 
Since that seems to be false, let's go back a step.
Do you also use the definitions
[1, n] = \{ x \in \mathbb R \mid 1 \le x \le n \}
(for n \ge 1) and
N = \{ 1, 2, 3, 4, \ldots \}?
 

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