Solving Attached Problem, Plus MHF Update

  • Context: MHB 
  • Thread starter Thread starter Cairo
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around a mathematical problem presented by a participant, focusing on various parts of the problem that require proof or clarification. Participants explore different approaches to solving the problem, particularly parts (i), (iii), and the implications of convergence in sequences. There is also a side discussion regarding the status of the MHF website.

Discussion Character

  • Homework-related
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant seeks help with an attached problem, indicating progress on some parts but confusion on others, particularly part (i).
  • Another participant questions the absence of the attachment and provides a link for further information about the MHF website.
  • Some participants propose that for part (i), if $a=0$, the result is clear, while for $r<\infty$, the convergence of the sequence $\{a_n\}$ to $0$ can be used.
  • It is noted that for $r=\infty$, the proposed result does not hold, illustrated by the example $a_n=1-\frac{1}{n}$.
  • One participant expresses difficulty in translating hints into a proof, indicating a lack of clarity on how to demonstrate that a convergent sequence attains its supremum.
  • Another participant provides a sketch of proof, suggesting that if $a=0$, the result is obvious, and if $a\neq 0$, an integer $N$ can be found such that $|a_n|\leq \frac{\lVert a\rVert_{\infty}}{2}$ for sufficiently large $n$.
  • There is a repeated emphasis on the participant's struggle to understand the transition from convergence to attaining the supremum, questioning the rationale behind specific inequalities used in the proofs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof for part (i) of the problem, with some providing insights while others express confusion. The discussion remains unresolved regarding the best approach to demonstrate the required result.

Contextual Notes

There are limitations in the clarity of the problem due to the missing attachment, which may affect the understanding of the context. Additionally, the discussion includes unresolved mathematical steps and assumptions regarding the properties of convergent sequences.

Who May Find This Useful

This discussion may be useful for students or individuals working on mathematical proofs related to sequences and convergence, particularly those seeking collaborative problem-solving approaches in a forum setting.

Cairo
Messages
61
Reaction score
0
Could anybody help me with the attached problem?

I've managed to prove part (ii) and (iv) of the question, but have no idea where to begin with part (i). I'm assuming that the result follows by using the definitions, but cannot make any progress with it.

For part (iii) I have thought about looking at the ration of the two norms and then using the squeeze rule but again, have yet to make any progress with this.

Any help would be much appreciated here.

Do we also know when MHF will be up and running again?
 

Attachments

Last edited:
Physics news on Phys.org
Where is the attached problem?
 
dray said:
Do we also know when MHF will be up and running again?

I don't see any attachment and please read http://www.mathhelpboards.com/showthread.php?119-is-the-mhf-dead in regards to MHF. This is a new site that will remain for a long time. Shoot me a PM if you can't work out the attachment issue.
 
For question i), if $a=0$ it's clear, and otherwise, for $r<\infty$ use the fact that the sequence $\{a_n\}$ converges to $0$. For $r=\infty$, it's not true (take $a_n=1-\frac 1n$).
 
girdav said:
For question i), if $a=0$ it's clear, and otherwise, for $r<\infty$ use the fact that the sequence $\{a_n\}$ converges to $0$. For $r=\infty$, it's not true (take $a_n=1-\frac 1n$).

Thanks for replying so quickly, Girdav. However, I'm still not sure how to translate this into a proof as such.
 
That's your job, so I will only give a sketch of proof. If $a=0$ then the result is obvious, and if $a\neq 0$, since $\lim_{n\to \infty}a_n=0$ we can find an integer $N$ such that $|a_n|\leq \frac{\lVert a\rVert_{\infty}}2$ if $n\geq N$. So the index we are looking for is between $0$ and $N-1$.
 
I've tried and tried and tried and still don't get this!
 
What do you get?
 
dray said:
Do we also know when MHF will be up and running again?

Nobody knows. What I do know is that MHB is not going away any time soon. It also does not suffer from the same problem MHF did: a site owner who had orphaned the site.
 
  • #10
girdav said:
What do you get?

It's just the first part I am unsure of. Am I to show that a convergent sequence attains its supremum?
 
  • #11
Yes, and here it's a little more easy since it converges to $0$. Did you try with my hints?
 
  • #12
girdav said:
That's your job, so I will only give a sketch of proof. If $a=0$ then the result is obvious, and if $a\neq 0$, since $\lim_{n\to \infty}a_n=0$ we can find an integer $N$ such that $|a_n|\leq \frac{\lVert a\rVert_{\infty}}2$ if $n\geq N$. So the index we are looking for is between $0$ and $N-1$.

OK - I give up with this problem! I just cannot see how you make the jump from converges to zero on to attaining its supremum.

Where did the 2 in the denominator come from in your inequality?

I'm at least partially happy that I have managed to prove the remaining parts of the problem.
 

Similar threads

MHB How can we prove the inequality for the supremum and infimum of f*g and f*g?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Uniform norm .... Garling Section 11.2 Normed Saces .... also Example 11.5.7
  • · Replies 4 ·
Replies
4
Views
2K
MHB Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3
  • · Replies 2 ·
Replies
2
Views
2K
Poisson random process problem
  • · Replies 10 ·
Replies
10
Views
2K
MHB Proof of the Cauchy-Schwarz Iequality .... Garling, Proposition 11.3.1 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (c) ....
  • · Replies 3 ·
Replies
3
Views
2K
MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (b) ....
  • · Replies 5 ·
Replies
5
Views
2K
MHB Limit of a Sequence (Updated with progress)
  • · Replies 2 ·
Replies
2
Views
1K
MHB Residue theorem to evaluate integrals
  • · Replies 16 ·
Replies
16
Views
3K
Average velocity for two different types of motion along same path
  • · Replies 22 ·
Replies
22
Views
1K