Proof of limit inequality
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
Where's the Strangedupe blog post? Previously, in order to explain the benefits of (certain limited kinds of) duplication on Stack Exchange sites, I've seen a blog.stackoverflow blogpost with the words "Dr. Strangedupe" in the title.  The post comes up in a search on the new blog, but       Can we please bring this post back? I'd love to link to it when , to reassure the questioner that there is value in not deleting their question because of search terms, but the blog post explained it a lot better than I could in a comment. C++ Program communicating with MySQL Database Does anyone know of a simple way for a C++ program to communicate directly with a MySQL database?  I have looked at MySQL++ and found it to be very confusing.  If anyone knows of a very simple way of doing this, please let me know.  Thanks  P.S. I am developing on a Windows machine.  PHP and MySQL web web application setup.  C++ setup to talk to the serial port.  Thus why I need a way for C++ to talk to the MySQL database directly. Schengen refusal formulae can be difficult to understand sometimes.  Sometimes an applicant receives a refusal with:       Justification for the purpose and conditions of the intended stay was not reliable.     often accompanied by:       Your intention to leave the territory of the Member States before the expiry of the visa applied for could not be ascertained.   What is meant by this refusal reason?  What are they trying to say? Isomorphism equivalence implies fundamental group is abelian Under what conditions will healers switch targets? I've been using healers lately, which I haven't really utilized much before. I've noticed that they will sometimes switch targets, even when their original target still lives.   What conditions are needed for healers to switch targets? Why does the Community user approve and reject edits? "Speed" of Gravity and Speed of Light Some threads here touching speed of gravity made me think about that. This lead to some questions.    The speed of gravity was not measured until today (at least there are no undebated papers to that effect). It seems common knowledge/belief among physicists that the speed of gravity is the same as the speed of light.    And this is my question: Is that speed of light = speed of gravity a result of GR or is that fed into the theory? Or is there some evidence from other sources than GR for this? Does the same speed imply some deep-lying relation between gravity and electromagnetism? Assume that given three predicates are presented below: $H(x)$: $x$ is a horse $A(x)$: $x$ is an animal $T(x,y)$: $x$ is a tail of $y$ Then, translate the following inference into an inference using predicate logic expressions and prove whether inference is valid or not (for instance, using natural deduction): Horses are animals.  Horses' tails are tails of animals. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. My initial idea was to consider similar sentence such as &quot;w is a tail of a horse&quot; to form required inference, but it was not successful. Would be welcomed to hear your ideas about this task. This is a follow-on from my  about samples from a distribution.  Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that $P(X_n &gt; X_1 \wedge \ldots \wedge X_n &gt; X_{n-1}) = \frac{1}{n}$ (this is true intuitively and experimentally). In my online brokerage account, I want to buy a particular stock and I see the following:  Bid:  13.20 x200           Ask:  13.27 x1,000   the current stock price is 13.22.   Can someone explain what the bid and ask prices mean relative to the current price?  If I buy 1000 shares, why would I pay more? Is there a word in English to qualify the highest-ranked student in a class in a year?  We are talking here about higher education, especially in the context of curricula that keep a given set of students over the course of several years, being ranked at the end of each year, possibly with competitive admission for the following year. The ranking is a weighted average of all marks obtained over the year.  I am looking for a single word, or expression, to quickly designate the one student with the highest ranking. If there is no such thing in English, then let's try to get the most concise and idiomatic definition.  In all cases, it is not emotionally connoted, i.e. not pejorative nor bragging.  For French speakers: I am looking for the translation of “major de promotion”.    EDIT: additional, not as much of a pure “word request” question: is this anything you should mention on a curriculum vitæ in the English-speaking world? If so, please mention whether the given word is appropriate for such a use&nbsp;&nbsp;:) I'm using Ubuntu 12.04 &amp; everything about it is great except that it just won't resume after suspend. I've tried the suggestions listed in this thread  but none worked. I'm using Lenovo Y510P. I'm new to Ubuntu, or unix for that matter, so please let me know if there's anymore  information that I should be posting. Thanks.  update  I tried the suggestion by Dima on this post . The machine wakes up from hibernation but immediately the screen gives a screen like this     Wondering if this is an issue with the graphics card or something. What is the monomorphism restriction? I'm unable to get Atom text editor from the Software Center. When I try I get this error:  Package does not exist. Check the spelling of the package name, and that the appropriate repository is enabled. What is the procedure followed by game programmers for saving and resuming game state? like files/mappings. I want to save it for tower defense game and I am using unity3D game engine. Run VirtualBox in background, without a window? What is the command to remove all files but not directories? Let's say I have a directory tree like this:  FOLDER:     file1     file2     file3     Subfolder1:         file1         file2     Subfolder2:         file1         file2   If I used rm -r FOLDER/*, everything in FOLDER would be deleted including sub-directories. How can I delete all files in FOLDER and in its sub-directories without deleting actual directories? Why are there two different hats for "Where in the World"? I am wondering why some got a moon and others got a sun as Where in the World hat for the same action:   vs.   And is it possible two collect both e.g. from different sites? I only get moons…  I know that there has already been some spoilering . We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2? How to run scripts on start up?