True or False? Sequence Question for {sn+tn}

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Homework Help Overview

The discussion revolves around the properties of sequences, specifically focusing on the behavior of the sum of two sequences, {sn} and {tn}, where {sn} is oscillating and not bounded, while {tn} is bounded. The original poster questions whether the resulting sequence {sn+tn} remains oscillating and not bounded.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to find a counterexample to support their belief that the statement is true, while also seeking clarification on the definition of oscillating. Another participant explores the implications of the boundedness of {tn} on the sum {sn+tn} and presents a logical argument regarding boundedness.

Discussion Status

The discussion is active, with participants exploring definitions and logical implications. Some guidance has been offered regarding the properties of oscillating sequences, and there is an ongoing examination of the assumptions involved in the original statement.

Contextual Notes

There is a noted ambiguity in the definition of "oscillating," with different interpretations being discussed among participants. The original poster has expressed difficulty in proving their stance, indicating a potential gap in understanding the concepts involved.

happyg1
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Hi,
Here's the question:
If {sn} is oscillating and not bounded, and {tn} is bounded, then {sn+tn} is oscillating and not bounded.
True or False?

I have tried for a while to find a counterexample, but I can't. I am leaning towards saying that this is true...but then I have to prove it. Am I correct in saying that it's true?
I have also started trying to prove it, but I'm having a hard time. Any clarification or advice will be appreciated.
CC
 
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{sn} is not bounded, and let's say that it's not bounded above in particular. Let t be inf{tn}. Then (sn + tn) > (sn + t) for all n, so if {sn + t} is not bounded above, then neither is {sn + tn}. But if {sn + t} is bounded, then there is some K such that sn + t < K for all n, so sn < K - t for all n, so {sn} is bounded above, contradiction.
 
Hi,
Thanks for the help. I was trying to prove it directly and I wound up with a bunch of nonsense. I understand the problem now, and I have worked the other 3 that were giving me headaches.
Thanks
CC
 
What, precisely, do you mean by "oscillating"? I have seen it used to mean that terms alternate sign or simply to mean that sn-1< sn but sn> sn+1.
 
Our definition of oscillating is a sequence that diverges but not to + infinity or - infinity.
 

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