I know that $\sin(n)$ has a convergent subsequence, but is there any way to actually construct a convergent subsequence of this sequence, or does existence just follow from the Bolzano Weierstrass theorem, and a sequence cannot be constructed?
Define a sequence $(a_n)_{n = 1}^{\infty}$ of real numbers by $a_n = \sin(n)$. This sequence is bounded (by $\pm1$), and so by the , there exists a convergent subsequence.  My question is this: Is there any way to construct an explicit convergent subsequence?  Naïvely, I tried considering the subsequence $$\sin(3), \,\sin(31), \,\sin(314), \,\sin(3141),\, \sin(31415),\, \dots$$ hoping it would converge to $0$, but I was unable to prove this (and I think it's probably not even true).
Best Algebraic Geometry text book? (other than Hartshorne) Comment font sizes are inconsistent on Android An issue similar to that described in  and likely to be related.  The comments for a question may vary in size:         Chrome for Android on Nexus 7 running 4.2.2 Jelly Bean. This also occurs on Firefox for Android: Set a default parameter value for a JavaScript function By the year 1582 vs in 1582 1.In 1582, scientists pointed out that the calendar was 10 days behind the sun      2.By the year 1582, scientists pointed out that the calendar was 10 days behind the sun.   I wanted to know the difference between "by the year 1582" and "in 1582". I don't want to use the Super for that, I want to use Alt+Space instead. StringTokenizer? Convert the String to a char[] and iterate over that? Something else? How do I set a default host for nginx? I'm trying to figure out how to set a default host for my nginx installation. I found this article in the nginx Wiki:    Unfortunately, this doesn’t work. After restarting I get this:  Restarting nginx: nginx: [emerg] unknown directive "http" in /etc/nginx/sites-enabled/catchall:1 nginx: configuration file /etc/nginx/nginx.conf test failed   After removing the http directive I get this:  Restarting nginx: nginx: [emerg] unknown log format "main" in /etc/nginx/sites-enabled/catchall:7 nginx: configuration file /etc/nginx/nginx.conf test failed   I’m on Ubuntu 10.04.3 where I’m using the official nginx PPA. Version 1.0.9 of nginx is running. How can I create launcher icon for PhpStorm? Can I backup my entire Ubuntu installation to ISO DVD? Passing more then 100 arguments to ColdFusion function throws an error How to run Windows 10 from an external hard drive? I am currently able to boot from an external drive or multiboot USB. Taking this concept one step further, I have 3 questions:   1) Is it possible to install and run Windows 10 on a portable external hard drive or USB card with the same functionality of being installed on the internal drive?   2) Is it considered legal to do so or will it affect Windows updates in any way?   3) What are the steps to do that? (please outline steps I can follow so I don't lose a drive in the process). :)  EDIT: My question deals specifically with Windows 10, the legality and method of making it a standalone OS running from an external device and what effect if any it has on Windows Update; questions not dealt with in the other question. How can I count the number of records that have a unique value in a particular field in ROR? Drush from GitHub stopped working How do I change the launcher commands? XeTeX: What happened to \mathcal and \mathscr? Recently I switched from pdflatex to xelatex to be able to use OpenType fonts. I like to use variables highlighted using \mathcal for specific variables, while I used to use \mathscr-typeset variables for matrices. But when I compile with XeLaTeX they look exaclty the same.  Old code  \documentclass{standalone} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        New code  \documentclass{standalone} \usepackage{unicode-math} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Broken code  I tried to get back the glyphs from mathrsfs, but it failed this way.  \documentclass{standalone} \usepackage{unicode-math} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Question: How can I get back the behaviour? In principle I'd like to have the old \mathcal letters from the pdflatex-version of Computer/Latin Modern.    Alternate solutions (Addendum 2016/05/11) In Windows 8 I can reset everything and reinstall Windows. I assume this will clear my drive and start from scratch? Is that right? Does this create a Windows.old folder?  I see this dialog, but aren't my settings and preferences saved in the cloud? If I screw up my settings and reset like this, are my settings also refreshed back to defaults?  Since I use SkyDrive to sync my files, my personal files are safe, right?    Also, in Windows 8 I can refresh my PC without affecting files. Are my desktop and/or Windows store apps uninstalled when I refresh my PC?  I see this dialog, but I still am not sure. For example, when it says apps are "kept" does that mean I don't have to buy them again or are they installed after the refresh is over for me? I assume "from disk" means desktop apps? Or maybe corp apps?    What would motivate a user to choose between these two? (already output) in Debug logs I am trying to add a list into an another nested list before adding values of a list to nested list it is getting the correct value but after adding i am getting (already output) in debug log when i checked.    public List&lt;String&gt; fieldName {get;set;}  public List&lt;List&lt;String&gt;&gt; fileValuesXML; // To get the field values of xml  for(Dom.XMLNode child : SalesforceObjecet.getChildElements()) {                        objecetName = child.getText();             system.debug('Object Name in XML field  ' + objecetName );              for(Dom.XMLNode subchild : child.getChildElements())              {                 system.debug('Sub child ' + subchild.getText() );                 for(Dom.XMLNode subchildfield : subchild.getChildElements())                 {                     system.debug('Sub child Field ' + subchildfield );                     fName  = subchildfield.getText();                     fieldName.add(fName);                     //fileValuesXML.add(fieldName);                     //system.debug('Filed Name in XML field  ' + fieldName);                 }                  system.debug('Filed Name before affing to filevalueXML  ' + fieldName);                                     fileValuesXML.add(fieldName);                 fieldName.clear();                 system.debug('Filed Name after affing to filevalueXML  ' + fileValuesXML);             }    }    I am getting values in debug log as:     Filed Name before affing to filevalueXML  (AccountSource, field3)   Filed Name after affing to filevalueXML  ((), (already output),   (already output)) How to get the Vista taskbar to span multiple monitors? I can't for the life of me get the vista taskbar to span multiple monitors.  I've searched and looked far and wide, and alas the only pseudo-solution i can find is Ultra-Mon, but a) it doesn't quite look the same  and b) i don't want to have to pay for something that Windows really ought to be able to do out of the box.   Does anyone have a solution for this?  Will this get addressed in Windows 7? Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous Let $k:[0,1]\times[0,1]\rightarrow \mathbb{R}$ continuous and $K:C[0,1]\rightarrow C[0,1]$, given by $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$. Prove that $K$ is continuous. I try to see continuity in $0$, but i can't find $\delta$. Is $L^{p}$ space with alternate norm Banach?