Consider the sequence defined by $a_1 = \sqrt{2}$, $a_2 = \sqrt{2 + \sqrt{2}}$, so that in general, $a_n = \sqrt{2 + a_{n - 1}}$ for $n &gt; 1$. I know 2 is an upper bound of this sequence (I proved this by induction). Is there a way to show that this sequence converges to 2? What I think is that the key step is to prove 2 is the least upper bound of this sequence. But how?
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
Why doesn't Java allow generic subclasses of Throwable? According to the , 3rd edition:        I wish to understand why this decision has been made. What's wrong with generic exceptions?  (As far as I know, generics are simply compile-time syntactic sugar, and they will be translated to Object anyway in the .class files, so effectively declaring a generic class is as if everything in it was an Object. Please correct me if I'm wrong.) SR Flip-Flop: NOR or NAND? Why are there holes in the icing? Is there any hint that the MoM or anywhere else got support in combating voldemort the first time around? Or is everything completely UK-centric?  I can't remember anyone in the second war being specifically sent from europe to help, even though we see international wizards in book 4. But I don't think all the events of the first war are covered in the books.  So are other bodies like the MoM involved in the first war, or was it isolated in the UK? Easiest way to import access db (.accdb) into SQL Server 2016? I have a .accfb file from an access 2016 database and am trying to import this database into a SQL Server 2016 developer edition instance.   While I found posts online discussing using import/export data tools in SQL Server and also migration assistant for access, neither have worked out for me.   Does anyone know how to easily move a database from access 2016 to sql server 2016? More specifically an "accdb" file type?  All suggestions are greatly appreciated! Incorrect bookmarks and page number in table of contents I have problem with table of contents and with bookmarks. In pdf file   table of contents shows wrong page number of Introduction and Bibliography, Bookmarks: introduction and bibliography don't bring me to the correct page.   Part of my code  \documentclass[12pt,a4paper]{book}  \usepackage[polish]{babel} \usepackage{polski} \usepackage[cp1250]{inputenc} \usepackage[pdftex,pagebackref=false,draft=false,pdfpagelabels=false,pdfstartview=FitH,pdfstartpage=1,bookmarks=true,pdfauthor={LiN},pdftitle={PD},pdfsubject={Titlei},pdfkeywords={words},unicode=true]{hyperref}  \usepackage{tocloft}  \begin{document} \tableofcontents \addcontentsline{toc}{chapter}{Introduction} \chapter*{Introduction} \chapter{Chapter1} \section{Section} \addcontentsline{toc}{chapter}{Bibliography} \begin{thebibliography}{9999999}% \bibitem{1}Author, \emph{Title of book}, Publisher. \end{thebibliography} \end{document}   How can I change that? How do I choose between "as to", "on", "of", and "about"? What's the purpose of cycling gloves? I haven't cycled for many years already and long ago when I was cycling I didn't pay much attention to equipment. Now I see that many cyclists wear a certain kind of gloves - that only cover the palm and leave the fingers uncovered.  What's the purpose of wearing such gloves (and not usual gloves for example)? In what non-cycling activities can such gloves be useful? Let me start this with : Very long "About me" text squeezes both the left side (flair) and the right side (the section with location, GitHub link, etc.)  Now, the reverse thing has happened:         put a long text in their "location", and it causes the right section to be overly extended, effectively squeezing the About Me section.  When will the squeezing game end? What's the right way to hard boil eggs? Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance. I have some files on my Unix machine that start with    --   e.g. --testings.html  If I try to remove it I get the following error:  cb0$ rm --testings.html rm: illegal option -- - usage: rm [-f | -i] [-dPRrvW] file ...        unlink file   I tried   rm "--testings.html" || rm '--testings.html'    but nothing works.  How can I remove such files on terminal? Counting homomorphisms between $\Bbb Z$ and $\Bbb Z_n$ and also between $\Bbb Z_n$ and $\Bbb Q$ Is it Possible to Disable Feed and Topics for an Object Completely at Profile Level / Page LayOuts Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$. How do I add a border around windows but keep the current theme? I seem to live my life in the terminal - many terminals in-fact.  I also love the Ambiance theme - but the new borderless windows does not provide the visual differentiation I'm used to (the same black windows appear to blend into each other).    Thus to my question - how do I add back the single pixel border for windows that is found in earlier Ubuntu versions?  I want to keep the Ambiance overall theme since this looks good and I dont want to swap to another theme.      Now I know that I can modify the Ubuntu Ambiance theme as follows:  mkdir ~/.themes cp -r /usr/share/themes/Ambiance .themes   Then flipping between the theme options in the Settings - Appearance picks up changes in the theme in now in the .themes folder.  I presume I need to alter one of the .css files but I dont know which one.  I was also looking at  but the solutions there doesnt work for 14.04.  I also looked at the  but got rather lost trying to figure out which one applies to my situation. Render just part of a final image from Cycles? What is $\Delta t$ in the time-energy uncertainty principle? So I currently have this issue where my Epson WF-3620DWF can't find the blue and yellow cartridges. (They are official epson ones) and I have a full black cartridge in there. How do I force my printer to ONLY print in black and ignore the cartridge(s) not recognised error? Should flour be thrown out past its expiry date?