I know the limit is $\sqrt 3$. And I tried proving that the sequence is decreasing from $a(2)$, yet when I got to this point, I have been stuck: $a(n+1) - a(n) = (3 - a(n)^2) / 2a(n)$ ? Thank you very much!
Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help.
Although Ubuntu 16.04 release notes mention that:  The default VIM package has been built against python3 instead of python2.  Yet after I installed vim with the following command:  sudo apt-get install vim   And checked:  vim --version   It showed:  -python -python3   Is it a bug? How do I get vim with Python support? Views indicator inaccurate, shows one views on question with two close votes Two close votes and only one person has viewed the question?    How did this happen? I want to install a portable Tex Live in windows 10. But after searching online I just get this from .  The -portable installer option (or V command in the text installer or corresponding GUI option) creates a completely self-contained TEX Live installation under a common root and forgoes system integration. You can create such an installation directly on a USB stick, or copy it to a USB stick afterwards. To run TEX using this portable installation, you need to add the appropriate binary directory to the search path during your terminal session, as usual. On Windows, you can double-click tl-tray-menu at the root of the installation and create a temporary ‘tray menu’ offering a choice of a few common tasks, as shown in this . The ‘More…’ entry explains how you can customize this menu.  Any more instructions in detail? What should I do? How well does jQuery support backward compatibility? We have had issues with Mootools not being very backward compatible specifically in the area of drag and drop functionality. I was wondering if anyone has had any similar problems with jQuery not being backward compatible. We are starting to use it quite heavily and are thinking about upgrading to a newer version to start using several plugins that require it. Will we have any issues if we get rid of the older version? Force ssh to not print "Remote host identification has changed" warning Is there a way to avoid ssh printing warning messages like this?   @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @    WARNING: REMOTE HOST IDENTIFICATION HAS CHANGED!     @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@   Although the remote host identity has changed but I know it is fine and  just want to get rid of this warning. How does one install PlayOnLinux? How do I install PlayOnLinux on Ubuntu? Difference between "try to do" and "try and do" What is the difference between try to do and try and do?  To me (non-native speaker), asking someone try and do this seems a bit rude. It's like saying you can try all you want but this must be done: try and fail is not an option. However, asking someone try to do this means I am asking you to try: success is bonus, failure is OK. I want to compile my program with the latest version of gcc.  Ubuntu 14.04 comes with gcc 4.8.2, however there's 4.9.0 available, moreover, I see that it is available as a package: . I tried to install it  sudo apt-get install gcc-4.9   but it says  Reading package lists... Done Building dependency tree        Reading state information... Done Note, selecting 'gcc-4.9-base' for regex 'gcc-4.9' gcc-4.9-base is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   Looks like it is already installed, just not as the default one? How do I utilize it to build my program? Where's my Vox Populi badge? Possible Duplicate:          I was a very good boy yesterday, voting 40 times (and getting the message that told me to wait until the next day before voting again) on a good mix of questions and answers. But I didn't receive the Vox Populi badge for it. What gives? Which method performs better: .Any() vs .Count() > 0? in the System.Linq namespace, we can now extend our  to have the  and  extension methods.  I was told recently that if i want to check that a collection contains 1 or more items inside it, I should use the .Any() extension method instead of the .Count() &gt; 0 extension method because the .Count() extension method has to iterate through all the items.  Secondly, some collections have a property (not an extension method) that is Count or Length. Would it be better to use those, instead of .Any() or .Count() ?  yea / nae ? Flag Question Link After 3k Rep I noticed from a question about flagging vs closing (forgive me, I cannot find the link, please post it if you know it), that flags are CONVERTED into close votes for questions automatically. Can somebody confirm this?  If confirmed, I was wondering why the link (again, for questions only) is simply not removed to avoid confusion? Is it possible to install an earlier version of Skype on Windows 8? Verify mysqldump backup is corruption free Why does triply periodic univariate function not exist? A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if   $$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$$   for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function.  that a triply periodic univariate function cannot exist, but why? Can't we follow the construction of elliptic functions that uses Schwarz-Christoffel formula to map upper-plane onto a triangle, and use the reflection principle to create a lattice which corresponds to three periods?     Which step fails to hold when we mimic the construction of an elliptic function to create a triply period meromorphic univariate function?   Sorry if the question is stupid. I do not have an access to the original proof that a triply periodic function cannot exist, either. So references are welcome! Continuous function positive at a point is positive in a neighborhood of that point Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) &gt; 0$ then there exists a $d &gt; 0$ such that $\forall x$, $f(x) &gt; 0$ with $|x-c| &lt; d$.   I can understand what the problem means. That if a function is positive at a point then there is another point that's really close that'll also be positive. I can not prove this in a formal way though. I've tried using the intermediate value theorem and I do not know hot to implement it. Any ideas? Dedicated mysql server slow response How to list all functions in a Python module? I have a Python module installed on my system and I'd like to be able to see what functions/classes/methods are available in it. I want to call the help function on each one. In Ruby I can do something like ClassName.methods to get a list of all the methods available on that class. Is there something similar in Python? eg. something like: from somemodule import foo print(foo.methods)  # or whatever is the correct method to call How to pass parameters to an alias? For bash script, I can use "$@" to access arguments. What's the equivalent when I use an alias? "Never say die?" What's the deal with this idiom? I know what it means, but cannot wrap my head around its grammar structure. The sentence structure of     "Never say die"   looks like that of     "Never admit defeat"   but it does not make the same grammatical+semantical sense.  Is it an archaic use? Is it a contraction of a longer phrase? Was it originally misspoken by someone of note and survived memetically? What are its roots?  Edit: Well, the comments made me realize that I worded my question poorly. I'm not saying I think this phrase is ungrammatical. I'm asking how the heck "saying 'die'" came to mean what it means. As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click).