Stupid questions of basic analysis

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Discussion Overview

The discussion revolves around the relationship between the limit of a sequence and the infimum of its values. Participants explore whether the condition \(\lim t_n > -\infty\) implies that \(\inf\{t_n:n\in\mathbb{N}\} > -\infty\), examining the implications of boundedness and the definitions involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that if \(\{t_n\}\) takes values in \(\mathbb{R}\) and has a limit, then the infimum should be greater than \(-\infty\), suggesting an if-and-only-if relationship.
  • Another participant challenges this by stating that the existence of a limit does not guarantee that the infimum is a minimum, emphasizing that a sequence with a limit is bounded.
  • This participant provides a sketch of the proof, indicating that only finitely many points can be far from the limit, which affects the determination of the lower bound.
  • A later reply acknowledges a mistake in assuming that the bound can be attained within the finite set in the infimum, recognizing that this is not necessarily true.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the implications of the limit on the infimum, and the discussion remains unresolved.

Contextual Notes

Participants express uncertainty regarding the definitions of infimum and minimum, and the implications of boundedness in relation to limits. The discussion highlights the need for clarity on these concepts.

jessicaw
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Why [itex]\lim t_n >-\infty\Rightarrow inf\{t_n:n\in\mathbb N\} >-\infty[/itex]?
 
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I think the argument is, first of all, I assume [tex]\{t_{n}\}[/tex] take values in [tex]\mathbb{R}[/tex], then, due to the existence of limit, [tex]\inf[/tex] is indeed [tex]\min[/tex] and so it should be [tex]> - \infty[/tex]. Somehow I think it is also an if-and-only-if statement.

Wayne
 
The existence of the limit does not imply infimum is minimum.

It's a general fact that if a sequence of points has a limit, the sequence is bounded. The proof can be sketched as follows: Only finitely many points can be a distance greater than 1 away from the limit (by the definition of a limit). So a lower bound of the set is either one of the values farther away than 1 from the limit, or one less than the limit is a lower bound
 
Argh, you are right.

I made a mistake in assuming that the bound can be attained within the finite set in the infimum but indeed it is not necessary true. Thanks.

Wayne
 

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