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How does one prove that every finite group is isomorphic to a subgroup of an alternating group? Moderators accepting answers on user's behalf after a certain time period Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$ Prove that sequence $$ x_{(n+1)}= \frac {a}{1+x_n}$$ is convergent to positive root of $x^2+x-a=0,$where $a &gt;0$ and $x_1 &gt;0$  we have $$x_{(n+2)}-x_n=\frac{-a(x_{(n+1)}-x_{(n-1)})}{(1+x_{(n-1)})(1+x_{(n+1)})}$$  Hence from this we see behaviour of even terms subsequence and odd terms subsequence  is opposite. how to proceed from there. any hint please I want to use the top half of the circle to make a cut but whenever I apply the boolean modifier, the circle just stays there and becomes attached to my other object.  I've managed to make the cut in a standard cube but not sure why it's not working for this...? Every retraction is a quotient map? I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$  is a continuous map s.t. $r(a)=a$ for every $a\in A$. The word "Print" appears after every source using \printbibliography Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain.    Morally, all we need to do is find some nonzero element that can be written in two distinct ways as a product of irreducible elements, but nothing is coming to mind immediately. Thoughts? I am looking for an animated series, shown in the 90s. I remember an episode where the boy went into VR and fought a medusa-like monster.  It may be that this was the premise for the whole series: That the children heroes go into VR every time to fight the evil guys.   I am not sure if the whole series were computer-animated but surely there were such elements together with classic animation.  The style of the series was quite dark and cool for teenage boys. Automatically keep current version of config files when apt-get install Optimization paramters read-only mysql 5.7 database? What's the best way to configure a MySQL database that is only read-only, and will never have any modifications made to it?  1) I have increased my innodb_buffer_pool_size to 70% of available RAM  2) Increase query cache size to 40mb  Running against a load generator, after some time the database starts bottlenecking to only 1000 statements per second.  The configuration is 2-cores, 8GB ram 50GB SSD. The database contains a single InnoDB 13 GB table with the same exact query running on two indexed columns.  We are using a vertx server to hit the database. The symptom we see is that the web service performance degrades after 15-20 minutes of heavy load.  Where can the bottleneck be? What parameters should we optimize? What does the notation $\Gamma \vDash \phi$ mean (in Mathematical Logic)? What does the notation $\Gamma \vDash \phi$ mean (in Mathematical Logic)?  I understand that $\Gamma$ represents a system of formulas, and that $\phi$ represents an individual formula.  I also know that $\Gamma \vDash \phi$ means " $\phi$ is a semantic consequence of $\Gamma$" - but I do not understand what this actually means. Can someone explain this idea to me in layman's terms?  Furthermore, suppose I had the following question:     Let $\Gamma \cup \{ \phi \}$. Does $\Gamma \vDash \phi$ hold for the set of formulae   $$ \Gamma = \{ p \rightarrow q, q \rightarrow r, r \rightarrow s \} \hspace{10 mm} \text{ where } \phi \text{ is } p \rightarrow s $$   Would I attempt this question by trying to prove that, for every formula $\psi \in \Gamma$, the valuation $v(\psi) = T$, based on the assumption that $v(\phi) = T$? Why can't I catch Pokemon in Pokemon Go anymore? Horizontal line of custom length like an — Java Socket why server can not reply client Why is there no overload of Interlocked.Add that accepts Doubles as parameters? How do I make Panel always transparent like the Dock in GNOME 3? I'm using Ubuntu 17.10. I want the Panel to be always transparent instead of dynamic transparency. What is the module/method used to get the current time? BibTeX bibliography not printing I was unable to find a previous post about this - perhaps too simple? I just came across the BibTeX and am trying to use it with an EndNote library. I have exported to the .bib file (which looks good as far as I can tell), but no bibliography is printed and the references aren't being cited. Obviously there is something major missing.  My document looks as follows.  \documentclass[11pt, fceqn]{article} \usepackage{epsfig} \usepackage{setspace, titlesec, caption, floatrow}    \usepackage{amsmath, amssymb, amsfonts, amsthm, cite, mathtools, lineno, subfig, natbib}  \begin{document}   Some text \cite{Alo_1}.  \bibliography{bib} \bibliographystyle{plain}  \end{document}   Any thoughts would be greatly appreciated. Difference In Eigenvectors Oxford Comma Conventions