- #1

Quantumpencil

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## Homework Statement

Is it true that if {pn}{qn}-> 0, that either pn or qn converges to 0?

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- Thread starter Quantumpencil
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- #1

Quantumpencil

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Is it true that if {pn}{qn}-> 0, that either pn or qn converges to 0?

- #2

Citan Uzuki

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- #3

Quantumpencil

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- #4

Quantumpencil

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(1,0,1,0,1...)

(0,1,0,1,0...)

My concern isn't relevant because they diverge.

- #5

HallsofIvy

Science Advisor

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But you can find sequences that do NOT converge but {x_ny_n} converges to 0.

- #6

Quantumpencil

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Would that be possible, or would {x_n}-> x imply {y_n} also converged as well?

- #7

Dick

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n*(1/n) converges, right?

- #8

Citan Uzuki

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Would that be possible, or would {x_n}-> x imply {y_n} also converged as well?

As dick showed, it is possible to have {x_n*y_n} and {x_n} converge, but {y_n} diverge. However, this is only possible if [itex]x_n \rightarrow 0[/itex]. Because if [itex]x_n y_n \rightarrow L[/itex] and [itex]x_n \rightarrow x \neq 0[/itex], then applying the main limit theorem shows that:

[tex]\lim y_n = \lim \frac{x_n y_n}{x_n} = \frac{\lim x_n y_n}{\lim x_n} = \frac{L}{x}[/tex]

i.e. [itex]y_n \rightarrow \frac{L}{x}[/itex]

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