Confused about the definition of a bounded sequence.

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Discussion Overview

The discussion revolves around the definition of a bounded sequence in the context of mathematical sequences. Participants explore the relationship between boundedness and the inclusion of sequences within closed and open intervals, examining the implications of these definitions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that for a sequence x(n) to be bounded, it must satisfy |x(n)| <= M, and questions how this relates to the definition involving closed intervals [a,b].
  • Another participant argues that if x(n) is in [a,b], then it is possible to choose an M such that b < M and -M < a, thus satisfying the boundedness condition.
  • A third participant adds that the relationship holds even if the interval [a,b] contains -M or M, or if -M < a, maintaining the boundedness claim.
  • One participant suggests that the concept extends to open intervals, asserting that if x(n) is in (a,b), it can still be bounded by choosing appropriate M values.
  • Another participant proposes taking M as the larger of |a| and |b| when x(n) is in (a,b), which would also satisfy the boundedness condition.

Areas of Agreement / Disagreement

Participants express differing views on the implications of boundedness in relation to closed and open intervals. While some argue that the definitions are equivalent, others highlight potential contradictions, indicating that the discussion remains unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the intervals and the values of M, as well as the implications of including or excluding endpoints in the intervals.

torquerotates
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Ok so for a sequence x(n) to be bounded it means |x(n)|<=M

but according to by book, if x(n) belongs to some closed interval, say [a,b], x(n) is bounded. That is confusing because say x(n) belonging to [a,b] means a<=x(n)<=b.

How can there exist a M such that -M<=x(n)<=M? this means that x(n) belongs to [-M,M]. If we take [a,b] to be an interval that doesn't contain 0, we get a contradiction.
 
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The two are equivalent. Say x(n) is in [a,b]. Take an M such that b<M and -M<a. Then x(n) is in [-M,M].
 
of course some interval [a,b] may contain -M or M, or if -M < a then it still holds.. et c
 
So I guess it also works for an open interval. so if all that we know about x(n) is that x(n) is in (a,b), we can claim that since there exist -M<a and M>b, |x(n)|<=M
 
If x in in (a, b), take M to be the larger of |a|, |b|. Then -M< x< M.
 

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