# Does the following argument work?

• Quantumpencil
In summary, the conversation discusses how to prove that if pn<qn for infinitely many n, then p<q. The conversation includes a lot of back-and-forth about rigor and the use of epsilon-delta proofs, but ultimately it is concluded that a simple proof by contradiction using the fact that if one sequence is smaller than another, the limits share the inequality is sufficient.
Quantumpencil

## Homework Statement

Suppose that a sequence {p}->p, and {q}->q. Show that if p_n < q_n for infinitely many n, then p<q.

## The Attempt at a Solution

I'm sure I know why this is true intuitively. The convergence guarantees that all but a finite number of p_n are within some distance of p, and all but a finite number of q_n are within some distance of q, for large enough n this distance is arbitrarily small, therefore, if an infinite number of p_n < q_n, then an infinite number of p_n < q_n when in some neighborhood of p which can be made as small as we like; hence in the limit, p<q.

Now, for actual rigor... Does it suffice to say that the d(p, p_n) < r, d(q, q_n) < r for some n if we take n > max (N, N') (where N and N' are the natural numbers such that the above inequalities hold in each case). Therefore in any interval around p and q, there are an infinite number of p<q, ect.

I know this should be an epsilon argument, but I don't know how to make it less "qualitative"

This is my problem with math. Urgh. Help please.

Edit: Yeah, I should've said < or =, but there is = on the keyboard, so I let that slide.

Last edited:
Actually this does not hold, we could also have p=q. For example consider the sequences q_n=1/n and p_n=-1/n.

If I would prove this, i would make a proof by contradiction. Assume that p>q, and then use the epsilon-delta definition for the limits to arrive at a contradiction.

Rigor: If a sequence converges to a point, every subsequence converges to that point.

For infinitely many n, we have pn<qn. We can thus take the sequence of all such n, which we'll label n(k) (I'm not going to use subscripts for that as double subscripting doesn't work out on the forum)

pn(k)<qn(k) for all k

What can you do with that?

You can then say that since the sub-sequence converges to p and q, p must be less than q, since every p_nk < q_nk and in the limit of large n they are approximately equal.

Still not rigorous enough though, I know.

Last edited:
Still not rigorous enough though, I know.

Wrong! You don't NEED to use epsilons and deltas. You should have proven: If an<bn for all n, and an converges to a, bn converges to b, then a is less than or equal to b (which you shouldn't forget the = part, as Kurret pointed out).

So pn(k)<qn(k) for all k, and hence p<=q. That's it No more work to be done

That is in fact substantially easier. Hm. I had heard from class-mates it was an Epsilon proof and wasn't able to get that to work out. All my triangle inequalities yielded useless information.

If it doesn't take too much effort, how would one do this using the given distance relationships?

The fastest way would be to just take the proof that if one sequence is smaller than another, the limits share the inequality, and apply it directly to the subsequence you have (replacing the arbitrary sequence that you have in the general proof with the specific one here). All you're doing is changing the names of the sequences

## 1. Does the following argument work?

This question is frequently asked because people want to know if the argument being presented is logical and convincing. The answer depends on the specific argument being discussed. It is important to carefully analyze the evidence and reasoning presented in the argument to determine if it is valid.

## 2. How do you determine if an argument is effective?

The effectiveness of an argument can be determined by evaluating its logical structure, evidence, and reasoning. A strong argument will have a clear and valid logical structure and will be supported by relevant and reliable evidence. It will also address potential counterarguments and provide a convincing conclusion.

## 3. Can you provide an example of a successful argument?

As a scientist, I cannot provide a specific example without context. However, a successful argument in science will typically present a clear hypothesis, use reliable and relevant data to support it, and address any potential alternative explanations. It will also follow the scientific method and be replicable by others.

## 4. What are common flaws in arguments?

There are many potential flaws in arguments, including logical fallacies, biased or insufficient evidence, and faulty reasoning. It is important to critically evaluate arguments to identify any flaws and determine if the argument is valid and convincing.

## 5. How can we improve our ability to make effective arguments?

To improve our ability to make effective arguments, we can practice critical thinking and logical reasoning skills. It is also important to gather and analyze reliable and relevant evidence, consider alternative viewpoints, and clearly communicate our arguments. Additionally, seeking feedback and being open to revising our arguments can help improve their effectiveness.

Replies
6
Views
6K
Replies
2
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
1
Views
5K
Replies
1
Views
1K
Replies
8
Views
2K
Replies
5
Views
3K
Replies
2
Views
953