Analysis Help: Lim SnTn is +Inf

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Homework Help Overview

The discussion revolves around the behavior of the product of two sequences, specifically analyzing the limit of the product \( \lim SnTn \) given that \( \lim Sn = +\infty \) and \( \lim Tn > 0 \). Participants are seeking to establish a proof for this statement.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the nature of sequences and their limits, with one attempting to outline a proof involving the selection of bounds for \( Tn \) and the implications of \( Sn \) approaching infinity. Others question the original poster's understanding and the need for a proof when a reference is available.

Discussion Status

The discussion is active, with participants exploring different aspects of the problem and engaging in a dialogue about the proof's requirements. Some guidance on the approach to proving the statement has been provided, but there is no explicit consensus on the proof's completeness or clarity.

Contextual Notes

One participant notes that the statement and proof are directly taken from a textbook, suggesting that the original poster may have access to the proof but is seeking further clarification on specific parts.

Zygotic Embryo
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Let (Sn) and (Tn) be sequences such that the lim Sn = +inf and lim Tn > 0

Then lim SnTn = + inf
 
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What is the question? And what work have you done on this problem?
 
A friend of mine, gave it to me.

I don't know where to start
 
its a statement

i need a proof
 
what doyou know about sequences, and multiplication (or division) of sequences?
 
is it

Let M > 0

Select a real number m so that 0 < m < limTn.

There exists an N1 such that:
n>N1 implies Tn>m

Since limSn=+inf there exists an N2 such that
n>N2 implies Sn>(M/m)

Set N = max{N1,N2}.

Then n>N imples SnTn>(M/m)*m = M
 
I thought I recognized the wording you used for the statement of this theorem and its proof and sure enough, it is taken word for word out of the book "Elementary Analysis: the Theory of Calculus" by Kenneth Ross (pg 50-51). Its stated with proof as theorem 9.9

If you have the proof in front of you, why are you asking if that's it? Is there some part of the proof you don't understand?
 

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