If $x_1&lt; x_2$ are real numbers, and $x_n = \frac{1}{2}(x_{n-2} + x_{n-1})$ for every $n &gt; 2$, how do I show $\lim x_n = \frac{1}{3}x_1 + \frac{2}{3}x_2$?
This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does.
How to prevent a coauthor from stealing the article? Please go through the following incidents to understand the problem.  Mr. A is a Ph.D. student of some institute who was working with Prof. B of another institute on a very interesting mathematical problem since last three years. Recently, they came across a significant solution to it. They wrote a research article to publish in a very good mathematics journal that accepts the online submission.  Prof. B wanted to submit it to the journal by himself. Mr. A agreed.   A few days ago, Mr. A wanted to know about the submission of their article via telephone. Prof. B informed that the article is submitted. But, Mr. A received any confirmation e-mail neither from the journal nor Prof. B forwarded the submission mail to him. Prof. B did not upload it in arXiv, which is a free preprint repository.  Mathematics journals used to take a long duration to review and publish an article. Prof. B told that submitting this article to arXiv will immediately attract community attention. Many researchers will start working in the new direction. As a result, Mr. A and Prof. B may lose their opportunity to work on that problems. Also, the article may be stolen.  Mr. A did not ask Prof. B to forward the submission confirmation mail to him. Prof. might think that Mr. A is suspecting him. It may affect the relationship. Also, Prof. is a very powerful person in the administration, who may prevent an opportunity for Mr. A to gain a job. Again from some of his friends, Mr. A came to know that Prof. had distributed related research problems to his collaborators.  As a consequence, Mr. A is thinking that Prof. has submitted this work in his own name and he has lost his credit as the first author of this article.  Now the question is: How Mr. A may save his credit and his work without hampering his relationship with Prof B? PostgreSQL, SELECT from max id how many distinct values does it have? I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.  The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem.   The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:   "The proof must be wrong."  "You cannot achieve such a result."  "You do not understand well the notion of ..."   My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes,  what should you do? Could 'cogito ergo sum' possibly be false? Flash/SWF Editor For Ubuntu 12.10 If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open ? I really don't know where/how to start...  PS : I precise that by "compact" I mean that it is hausdorff and that any recovering of $G$ by open sets has a finite sub-recovering (sorry if I wasn't unclear) Why can't I connect to Pokemon Go via mobile data? It connects to Pokemon Go via wi-fi perfectly fine, but when I try and connect to Pokemon Go via 3G it just gets stuck at the first bar on the loading screen with the eventual pop up saying "Failed to get player information from the server." My 3G connection works perfectly fine with every other app I have, and has been easily able to stream HD videos so I can't see connectivity with 3G being an issue.   It makes even less sense as once connected via wi-fi, the game easily continues to operate via mobile data once I leave the house. However once the app crashes (which it usually does after 15-30 minutes) I try to reload it on mobile data and it gets stuck on the loading screen. If I can find a local free wi-fi spot nearby I can re-connect easily, and then move back onto mobile data.  I've tried removing and reinstalling Pokemon Go and that didn't work. I also reset all my app preferences on my phone and that didn't work.  Not sure if this helps, my phone is a Huawei P8, Android version 5.0.1.  I signed up to Pokemon Go using my Google account. Why must I wait two days before accepting my own answer? I got this message today while trying to accept my answer on my own question.  Why must I wait this time, if I am sure that the answer I answered is the right one, just want to understand the point of view?  I faced that just today on Super user and didn't get this message before. How to show a group is cyclic? One question asking if $\mathbb{Z}^*_{21}$ is cyclic.  I know that the cyclic group must have a generator which can generate all of the elements within the group.  But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine if a group is a cyclic group? Why am I getting different results from two servers that located in the same destination network? There's two servers with the same network IP, and the only difference between them is the last two numbers like: 37.211.15.247 and 37.211.166.178. The Hosting Provider claims that the subnet mask for both networks is 255.254.0.0  When I try the command Tracert (Traceroute) to test the ping and to see what hops does it take to reach the destination server, I get different ping times and different forward path results which is strange because they are all located in the same company host.  Is there something wrong? Shouldn’t it take the same path too?  I have added the the full IP addresses to both servers as requested by some commenters. Well the two servers aren’t “International” anyways, so as far as I know, the only thing that comes to mind is the possibility that the two are configured on different subnetworks just like what . Though it’s suspicious.  UPDATE: After doing so much research, I found out the Subnet of these two hosts. They are on a /15 (255.254.0.0) subnet mask. Unfortunately, this is what that hosting provider claimed to me which is not true, and the true Subnet they apparently use is a /17 (255.255.128.0) Subnet mask. It turns out that they were definitely on different Subnets. That's why I got different traceroutes when I tested these two IPs 37.211.15.247 and 37.211.166.178  because they are clearly on different Subnets. Thank you everyone for these good answers, especially  his answer helped me a lot. Why couldn't Harry see Thestrals at the end of Goblet of Fire after the senseless and shocking death of Cedric Diggory?   At the end of Goblet of Fire, the carriages still appeared horseless to Harry:     Hermione turned away, smiling at the horseless carriages which were now trundling towards them up the drive, as Krum, looking surprised, but gratified, signed a fragment of   parchment for Ron.   Goblet of Fire - Page 629 - British Hardcover - Adult Edition   At the beginning of Order of the Phoenix, Harry sees the Thestrals for the first time:     Here stood the hundred or so horseless stagecoaches that always took the students above first year up to the castle. Harry glanced quickly at them, turned away to keep a lookout for Ron and Hermione, then did a double-take.   The coaches were no longer horseless. There were creatures standing between the carriage shafts.   Order of the Phoenix - Page 178 - British Hardcover - Adult Edition   We know that only those who have seen death can see Thestrals:     ‘Don’ worry, it won’ hurt yeh,’ said Hagrid patiently. ‘Righ’, now, who can tell me why some o’ yeh can see ’em an’ some can’t?’   Hermione raised her hand.   ‘Go on then,’ said Hagrid, beaming at her.   ‘The only people who can see Thestrals,’ she said, ‘are people who have seen death.’   ‘Tha’s exactly right,’ said Hagrid solemnly [...]   Order of the Phoenix - Page 394 - British Hardcover - Adult Edition   A month went by between Cedric's death and the end of the spring term -- is this enough time for Harry to have truly internalized Cedric's death? Goblet of Fire indicates that Harry was in shock following Cedric's murder, but was able to relate the story of Cedric's death to Mr. and Mrs. Diggory.     When he looked back, even a month later, Harry found he had few memories of the following days. It was as though he had been through too much to take in any more. The recollections he did have were very painful. The worst, perhaps, was the meeting with the Diggorys that took place the following morning.   [...]   ‘He suffered very little, then,’ she said, when Harry had told her how Cedric had died.'   Goblet of Fire - Page 621 - British Hardcover - Adult Edition   So, then, why couldn't Harry see Thestrals at the end of Goblet of Fire? If $E$ has measure zero, then does $E^2$ have measure zero? Currently I'm just working through measure theory just to see if its something I would like to take.  Unfortunately I am stuck on this problem from Carothers.     If $m^*(E)=0$, then $m^*(E^2)=0$.   Where $m^*$ denotes outer measure and $$E^2=\{x^2:x\in E\}.$$  I toyed with the idea that $I_k &lt; 1 \Rightarrow I^2_k &lt; I_k$.  However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need). Can anyone help me identify a story about kids with silver blood? Is there a tool in Ubuntu to automatically merge two images having a common strip? I need to scan a large image, which will not fit at once in the scanner. I was thinking about scanning it in two parts and hand-editing the result in Gimp to join them. But probably an automatic tool can do the same more quickly and better.  EDIT I think I may have not made my point clear. My difficulty is not aligning two images. My problem is that one has to compensate for the rotation or distortion introduced by the manual process of putting the sheet into the scanner, and an automatic tool should obtain more precise results at analyzing common patterns in the two images to rotate and align them seamlessly. Is Morocco currently safe to visit despite the Ebola outbreak? I'm travelling to Morocco over the Easter period, and while I am aware that the current Ebola outbreak is mainly in Guinea, Liberia and Mali, and would have to cross the Sahara to get to Morocco, is it wise for me to go to Morocco at this time?  What are some basic precautions I can take in the case that the Ebola virus travels to Morocco? What's the recommended way to move a VirtualBox VM to another computer? I use VirtualBox 4.1.x on my Ubuntu machine and I’ve set up several virtual machines. Since there are several ways one can move a virtual machine in VirtualBox to another computer, I was wondering which one is the recommended way:   Use the “Import/Export utility.” Copy the entire virtual machine folder, containing the .vdi and .vbox files. Clone the VDI using “Virtual Media Manager” and then recreate a VM on the target machine but using the cloned VDI as the hard disk.   I have successfully used the 1st method several times and it has always worked. The problem is that after exporting and importing, the disk image is transformed into VMDK and not VDI anymore!  The 2nd method is probably the easiest but I’m not sure that simply copying the files will work or not on the target machine. When searching about this method, I found some people had problems in which they had to edit the VirtualBox.xml file to solve it!  At last, there’s the 3rd method, but it requires the extra work of creating a VM similar to the original VM configuration, which is not desirable.  It’s clear from the above explanation that my desired method is the 2nd one, but I need expert advice on this if it works or not. I don't want any XML editing getting in my way!  What’s the best method of safely transferring my VM’s to another computer with VirtualBox? C programming: How to program for Unicode? What prerequisites are needed to do strict Unicode programming?  Does this imply that my code should not use char types anywhere and that functions need to be used that can deal with wint_t and wchar_t?  And what is the role played by multibyte character sequences in this scenario? The preimage of continuous function on a closed set is closed. Is doing two PhDs a good path? I am studying a combined bachelor of engineering (electrical) and bachelor of mathematics; I just started this year and will graduate in 2018. The reason why I am doing double degrees and not a single degree is because I love both electrical engineering and mathematics and I could not ignore any of them. So with this in mind, I am thinking of doing two PhDs when I graduate (one in electrical engineering and one in mathematics). Is this a good path or I should concentrate on only one of them?