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So far, I didn't have the chance to use either of these services. Before starting, I would like to hear your opinions. What are the advantages and disadvantages? Why should I choose one over the other? I am mainly interested in the technical side, but I would also love to hear comments also regarding the plans and prices. Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ 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. Currently, if you access a deleted question you will get a short message informing you that the question has been deleted. However, if you click on a direct link to a deleted answer or comment (which you can get via notifications), nothing is displayed.  This is naturally quite confusing as you will get the message about a new answer or comment via notifications, click on the link and simply see nothing.  Therefore, I propose to display some sort of message to the user, informing her/him that the answer or question has been deleted. Note that this is different from always showing a deleted answer/comment, as I specifically request it in case you access the specific URL.  One example for a comment would be:   One example for a deleted answer is:   This question is related to ,  but requests a feature to circumvent the problem altogether. Why did Terence Tao write Proposition 2.1.11 about mathematical induction in "Analysis I"? Trying to dual boot Ubuntu 14.10 with Windows 10, last one not detected Software to read a QR code? I saw a  reading a book, but I don't have a smart phone to check it.    So I just took a picture of it and saved my PC. And I downloaded libdecoderqr0, libdecodeqr-dev, libdecodeqr-examples for reading it. But I don't know how to use it.    What do I do next, or is there a nicer way to read QR codes? I have a Dell Inspiron 1525 laptop and I am planning on purchasing a mac mini, new version , I am tight on money for a while and want to use my current laptop screen as monitor for mac mini, I just want to know if it is at all possible to do, I have windows 7 and ubuntu 11.04 installed in my laptop, and I don't really care if need to wipe them out, or purchasing a monitos is the only option left, also I am planning to use it for coding and stuff, so VNC is probably not a good choice. I wish to have long and short forms of command line options invoked using my shell script.  I know that getopts can be used, but like in Perl, I have not been able to do the same with shell.  Any ideas on how this can be done, so that I can use options like:  ./shell.sh --copyfile abc.pl /tmp/ ./shell.sh -c abc.pl /tmp/   In the above, both the commands mean the same thing to my shell, but using getopts, I have not been able to implement these? Unable to Enter a Country If I have a connecting flight through a country I am unable to enter. Can I retrieve my bags to recheck for my connect? call function declared below Is it possible to call a function which is declared below in bash?  Example  if [ "$input" = "yes" ]; then     YES_FUNCTION elif [ "$input" = "no" ]; then     NO_FUNCTION else     exit 0; fi  YES_FUNCTION() {   .....   ..... }  NO_FUNCTION() {   .....   ..... } I was reading a website where common English sentence errors are written -     INCORRECT - He doesn't know to swim.      CORRECT - He doesn't know how to swim.   I know He doesn't know how to swim is a correct sentence. But why is the other one wrong, is my question?  I know that know is a transitive verb. So, the following are correct -   i) He knows swimming.  Swimming is a gerund. We can also use the infinitive form here - to swim  ii) He knows to swim. Zero visibility while editing a message in a chat room I have made some changes to a file which has been committed a few times as part of a group of files, but now want to reset/revert the changes on it back to a previous version.   I have done a git log along with a git diff to find the revision I need, but just have no idea how to get the file back to its former state in the past. It's often the case that I want to change name servers because my site is messed up.  Well, it takes a very long time for the name server to propagate.  Is there something in DNS, or registrar or anything I can do to speed that up.  Flushing my dns server using  and it rarely works I have a JavaScript object. Is there a built-in or accepted best practice way to get the length of this object? const myObject = new Object(); myObject[&quot;firstname&quot;] = &quot;Gareth&quot;; myObject[&quot;lastname&quot;] = &quot;Simpson&quot;; myObject[&quot;age&quot;] = 21; How can I split a beamer bibliography across two slides? Why was the Holdo manoeuvre not used earlier? This question is related to "The Last Jedi":  The maneuvre that vice admiral Holdo (with the purple hair) uses towards the end of the film seems remarkably effective.   It seems that it would have worked pretty well much earlier. Especially as two other ships were just allowed to run out of fuel and be destroyed.  It seems doing it earlier with different assets would have incurred less losses. Yet nobody seemed to consider it. Any in universe reason why? What torque for caliper brakes onto carbon-alloy forks? I've recently got a new bike with a . This comes with deep-drop calliper brakes (Shimano R451 Dual Pivot). The first time out I bumped one of the brakes and it twisted around the single attachment bolt and rubbed on the rim.  I managed to easily twist it back by hand and carry on, however I thought the bolt probably needs tightening. The question is: How tight can it be?  As the fork is part carbon I've so far only dared to take it up to 4Nm, but it still seems possible to twist the whole calliper around with a bump of the hand. Maybe that's normal? Perhaps it's easier to bump around because of the longer deep-drop lever. Should I be aiming to get it really secure?    I'm not sure whether the brake mount in the fork is all carbon, or some of it is alloy underneath, but either way I'd like to know what a sensible torque is. As far as I can see from the  there is no such specification. How can I find this out? How to avoid pagenumering with text in scrlttr2 Have a Letter in scrlttr2. I cannot avoid/exclude the text before the page number 'Seite' (see print-screen). The  recommends to add \pagenumbering{arabic}. But for my document, it has no effect.    %--------------------------------------------------------------------------- \documentclass%% %---------------------------------------------------------------------------   [fontsize=12pt,%%          Schriftgroesse %--------------------------------------------------------------------------- % Satzspiegel    paper=a4,%%               Papierformat    enlargefirstpage=on,%%    Erste Seite anders    pagenumber=footcenter,%%   Seitenzahl oben mittig %--------------------------------------------------------------------------- % Layout    headsepline=on,%%         Linie unter der Seitenzahl    parskip=half,%%           Abstand zwischen Absaetzen %--------------------------------------------------------------------------- % Formatierung    draft=on%%                Entwurfsmodus ]{scrlttr2} %--------------------------------------------------------------------------- \usepackage{german} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc}  %--------------------------------------------------------------------------- \begin{document} %--------------------------------------------------------------------------- % Briefstil und Position des Briefkopfs \LoadLetterOption{DIN} %% oder: DINmtext, SN, SNleft, KOMAold. \makeatletter \pagenumbering{arabic}  %--------------------------------------------------------------------------- % Seitenstil \pagestyle{plain}%% keine Header in der Kopfzeile %--------------------------------------------------------------------------- \begin{letter}{Herr\\                 Mustermann\\                Straße 1\\                12345 Stadt} %--------------------------------------------------------------------------- \setkomavar{subject}{Beispielschreiben} %--------------------------------------------------------------------------- \opening{Liebe Herr Mustermann,}  here is some Text.  \pagebreak  here is some Text. %--------------------------------------------------------------------------- \end{letter} %--------------------------------------------------------------------------- \end{document} %--------------------------------------------------------------------------- Why is Euler's Totient function always even? I want to prove why $\phi(n)$ is even for $n&gt;3$.  So far I am attempting to split this into 2 cases.  Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will always be even.  Case 2: $n$ is not a power of $2$. This is where I am unsure where to go. I figure I will end up using the fact that $\phi(n)$ is multiplicative, and I think I'll get a $(p-1)$ somewhere in the resulting product which will make the whole thing positive, as $p$ is prime implies $(p-1)$ is even.