What is the Difference Between Almost Upper Bounds and Upper Bounds in Calculus?

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Discussion Overview

The discussion revolves around the concepts of "almost upper bounds" and "upper bounds" in the context of calculus, specifically addressing definitions and examples related to bounded infinite sets. Participants explore the implications of these definitions and clarify their understanding through examples and counterexamples.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant defines an "almost upper bound" as a number x for a set A where only finitely many elements y in A satisfy y ≥ x, and questions the implications of this definition with a specific example.
  • Another participant questions whether x must be an element of A, suggesting that 1 could be an almost upper bound for the set (0,1) since there are no elements in A greater than or equal to 1.
  • A different participant agrees with the previous point, noting that any x ≥ 1 would qualify as an almost upper bound for (0,1) and that they are bounded below by 1.
  • One participant provides a contrasting example, stating that for a set like (0, 1) ∪ {2}, any number greater than or equal to 2 is an upper bound, while any number greater than or equal to 1 is an almost upper bound, highlighting the distinction in different contexts.
  • Another participant notes that for finite sets, all numbers are considered almost upper bounds, emphasizing the importance of the set being infinite in the original theorem.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of x being an element of A and the implications of the definitions provided. There is no consensus on the interpretation of "almost upper bounds" versus "upper bounds," and the discussion remains unresolved.

Contextual Notes

The discussion includes various interpretations of the definitions and examples that may depend on specific conditions or assumptions about the sets in question. The distinction between almost upper bounds and upper bounds is not universally agreed upon, leading to further exploration of the concepts.

jgens
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The author of my calculus book defines an "almost upper bound" as follows: A number x is an almost upper bound for the set A if there are only finitely many number y \in A with y \geq x.

He then asks the reader to prove that if A is a bounded infinite set, then the set B of all almost upper bounds for A is non-empty and bounded below. This seemed simple enough but I got confused when I thought about it in terms of a concrete example. Here are my thoughts . . .

Let A = (0,1) and x be an almost upper bound for A. Clearly x can be written in the form x = 1 - \varepsilon where 0 < \varepsilon < 1. Since x is an almost upper bound, there should only be finitely many numbers y \in A with y \geq x. However, since the infinite sequence of numbers 1-\varepsilon/2, 1-\varepsilon/3, 1-\varepsilon/4, . . . are all in A and greater than x this is a contradiction.

Could someone help me figure out where I went wrong in my thinking? Thanks.
 
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I was very confused.when i first came acrosse this.best tthing to do is try and remeber and "swat for yur exmas" it worked for me
 
Why must x be in A? Wouldn't 1 be an almost upper bound of that set, since there are 0 elements in A greater than or equal to 1?
 
Tiny-tim: This is from Spivak's Calculus

Thanks for the explanation guys! I guess I just wasn't thinking about this the right way.
 
For an interval like that, an "almost upper bound" is exactly the same as an upper bound.

If the set were, say (0, 1)\cup \{2\} any number larger than or equal to 2 would be an upper bound but any number larger than or equal to 1 would be an "almost upper bound". Both sets are non-empty and bounded below.

For a finite set finite set all numbers are "almost upper bounds"- which is why that theorem specifies that the set must be infinite.
 

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