# In search of a counter-example

• Quantumpencil
In summary, the statement "if {pn}{qn}-> 0, that either pn or qn converges to 0" is not necessarily true. It is possible for both sequences to diverge or for one sequence to converge while the other diverges. However, if the product of the sequences converges to 0 and one of the sequences converges, then the other sequence must also converge to 0.
Quantumpencil

## Homework Statement

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

## The Attempt at a Solution

So far I've made an argument for why it doesn't have to follow (Because we don't know that both sequences converge, and we could have two divergent sequences or a divergent sequence and a convergent sequence whose product converges, for example the alternating harmonic series), However I can't think of any particular counter-example where the product converges to 0.

Try to arrange it so that the even terms of {p_n} converge to zero, but {p_n} does not, and conversely, the odd terms of {q_n} converge to zero, but {q_n} does not.

That can't happen though can it? If {p_n}(even)-> 0, then the even terms form a subsequence, so wouldn't the sequence have to converge to 0 also?

nevermind, I got it I think.

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

My concern isn't relevant because they diverge.

Yes, if {x_n} converges to x and {y_n} converges to y, then {x_ny_n} must converge to xy. So if {x_n} and {y_n} converge AND {x_ny_n} converges to 0, either x or y must be 0.

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

Hm... what about if {x_ny_n} converges, but only {x_n} or {y_n} converges?

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

n*(1/n) converges, right?

Quantumpencil said:
Hm... what about if {x_ny_n} converges, but only {x_n} or {y_n} converges?

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 $x_n \rightarrow 0$. Because if $x_n y_n \rightarrow L$ and $x_n \rightarrow x \neq 0$, then applying the main limit theorem shows that:

$$\lim y_n = \lim \frac{x_n y_n}{x_n} = \frac{\lim x_n y_n}{\lim x_n} = \frac{L}{x}$$

i.e. $y_n \rightarrow \frac{L}{x}$

## 1. What is a counter-example?

A counter-example is an example or situation that goes against or disproves a proposed theory, hypothesis, or statement. It is used to show that something is not always true or that there are exceptions to a general rule.

## 2. Why is it important to search for counter-examples?

Searching for counter-examples is important in the scientific process because it helps to refine and improve theories and hypotheses. By finding instances that do not fit with a proposed idea, scientists can further investigate and modify their understanding of the topic.

## 3. How do scientists search for counter-examples?

Scientists can search for counter-examples through various methods such as experiments, observations, and data analysis. They may also review existing literature and studies to see if any previous findings contradict their proposed idea.

## 4. Can a counter-example completely disprove a theory?

It is possible for a counter-example to disprove a theory, but it is not always the case. In some situations, a counter-example may be due to a flaw in the initial hypothesis or the limitations of the experiment. Scientists must carefully evaluate the evidence before concluding that a theory is completely disproved.

## 5. Are counter-examples always negative or can they also support a theory?

Counter-examples can be both negative and positive. While a negative counter-example goes against a theory, a positive counter-example supports it. Positive counter-examples can provide evidence for the validity of a theory or help to strengthen its overall understanding.

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