I have to find $\limsup$ and $\liminf$ of $a_n=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor : n \in\mathbb{N}\}$   I suppose that I have to relate to subsequences  . First one is $a_{n_k}$ for all n that maintains $\sqrt n\in\mathbb {N}$ , so the limit is $0$.  Second  one $a_{n_j}$ includes the rest of the elements.I suppose it converges to 1, but dont know how to prove it.
I've got the following question:  $A = \{\sqrt{n} - \left\lfloor\sqrt{n}\right\rfloor : n \in \mathbb{N}\}$  Find inf A, sup A    inf A is no big deal, but I have a hard time proving sup A is 1  I've tried to prove that: $\sqrt{n^2 - 1} -\left\lfloor\sqrt{n^2-1}\right\rfloor &gt; \sqrt{n} -\left\lfloor\sqrt{n}\right\rfloor $  but couldn't find a way to do that
Its a .vbproj and looks like this  &lt;Project DefaultTargets="Build" xmlns="http://schemas.microsoft.com/developer/msbuild/2003"&gt;     &lt;PropertyGroup&gt;         &lt;ProjectGuid&gt;15a7ee82-9020-4fda-a7fb-85a61664692d&lt;/ProjectGuid&gt;   all i want to get is the ProjectGuid but it does not work when a namespace is there...   Dim xmlDoc As New XmlDocument()  Dim filePath As String = Path.Combine(mDirectory, name + "\" + name + ".vbproj")  xmlDoc.Load(filePath)  Dim value As Object = xmlDoc.SelectNodes("/Project/PropertyGroup/ProjectGuid")   what can i do to fix this? Undefined control sequence and \chapter error I’m getting an “undefined control sequence” error with the following code, using TeXstudio:  \documentclass[11pt]{article} \begin{document} \begin{center} \chapter{CHAPTER 6} \end{center} \end{document} Scenario: I have a custom button on a standard layout, which will call an apex class to do some validation on a record. If not validated, it needs to throw out some errors on that page. What are the best UX options to achieve this?  I am not too sure how to achieve this. Within a trigger, you can use the addError method to add the custom errors.  Can my Apex class reference the fields of a standard object? The ideal approach would be to do the following:  Contact.Email.addError('Please enter the right format here');   How can I achieve this via an apex class that the custom button will call?  What I have thought: Having a VF Page popup on the screen so that user can see all the errors, but that's not the best approach as UX is a big deal for this application  Thanks. Adding a Method to an Existing Object Instance Unable to lock the administration directory (/var/lib/dpkg/), are you root? I get this error in Lubuntu 12.04 when I try to update in terminal:  $ apt-get update E: Could not open lock file /var/lib/apt/lists/lock - open (13: Permission denied) E: Unable to lock directory /var/lib/apt/lists/ E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root? I'm playing around, trying to write some code to use the  APIs to shorten a URL.  After reading , I tried:     TRIM_API_URL = 'http://api.tr.im/api'    auth_handler = urllib2.HTTPBasicAuthHandler()    auth_handler.add_password(realm='tr.im',                              uri=TRIM_API_URL,                              user=USERNAME,                              passwd=PASSWORD)    opener = urllib2.build_opener(auth_handler)    urllib2.install_opener(opener)    response = urllib2.urlopen('%s/trim_simple?url=%s'                               % (TRIM_API_URL, url_to_trim))    url = response.read().strip()   response.code is 200 (I think it should be 202).  url is valid, but the basic HTTP authentication doesn't seem to have worked, because the shortened URL isn't in my list of URLs (at ).  After reading  I also tried:     TRIM_API_URL = 'api.tr.im/api'    password_mgr = urllib2.HTTPPasswordMgrWithDefaultRealm()    password_mgr.add_password(None, TRIM_API_URL, USERNAME, PASSWORD)    auth_handler = urllib2.HTTPBasicAuthHandler(password_mgr)    opener = urllib2.build_opener(auth_handler)    urllib2.install_opener(opener)    response = urllib2.urlopen('http://%s/trim_simple?url=%s'                               % (TRIM_API_URL, url_to_trim))    url = response.read().strip()   But I get the same results.  (response.code is 200 and url is valid, but not recorded in my account at .)  If I use query string parameters instead of basic HTTP authentication, like this:     TRIM_API_URL = 'http://api.tr.im/api'    response = urllib2.urlopen('%s/trim_simple?url=%s&amp;username=%s&amp;password=%s'                               % (TRIM_API_URL,                                  url_to_trim,                                  USERNAME,                                  PASSWORD))    url = response.read().strip()   ...then not only is url valid but it's recorded in my tr.im account. (Though response.code is still 200.)  There must be something wrong with my code though (and not tr.im's API), because  $ curl -u yacitus:xxxx http://api.tr.im/api/trim_url.json?url=http://www.google.co.uk   ...returns:  {"trimpath":"hfhb","reference":"nH45bftZDWOX0QpVojeDbOvPDnaRaJ","trimmed":"11\/03\/2009","destination":"http:\/\/www.google.co.uk\/","trim_path":"hfhb","domain":"google.co.uk","url":"http:\/\/tr.im\/hfhb","visits":0,"status":{"result":"OK","code":"200","message":"tr.im URL Added."},"date_time":"2009-03-11T10:15:35-04:00"}   ...and the URL does appear in my list of URLs on .  And if I run:  $ curl -u yacitus:xxxx http://api.tr.im/api/trim_url.json?url=http://www.google.co.uk   ...again, I get:  {"trimpath":"hfhb","reference":"nH45bftZDWOX0QpVojeDbOvPDnaRaJ","trimmed":"11\/03\/2009","destination":"http:\/\/www.google.co.uk\/","trim_path":"hfhb","domain":"google.co.uk","url":"http:\/\/tr.im\/hfhb","visits":0,"status":{"result":"OK","code":"201","message":"tr.im URL Already Created [yacitus]."},"date_time":"2009-03-11T10:15:35-04:00"}   Note code is 201, and message is "tr.im URL Already Created [yacitus]."  I must not be doing the basic HTTP authentication correctly (in either attempt).  Can you spot my problem?  Perhaps I should look and see what's being sent over the wire?  I've never done that before.  Are there Python APIs I can use (perhaps in pdb)?  Or is there another tool (preferably for Mac OS X) I can use? Break a loop animation (Make Cyclic) to start a new one I've used the 'Make Cyclic' to loop these keyframes (image below). So, I want to break the loop at a certain time to do a new animation (insert another kf to do a different thing than the loop does). Any help how to do this?    Thanks in advance! I'm looking for a word that describes a person who is extremely dismissive and unopen to ideas or comments. Dismissive  feeling or showing that something is unworthy of consideration   For example,  Tom: 'Get your table organised! It's so-' Jack: 'Shush, bring your comments somewhere else.' Is the total energy of the universe zero? Weird display on screen before swipe lock Samsung Galaxy s6 edge plus How do you get vim to display wrapped lines without inserting newlines? I'm getting back in touch with my inner (g)vim due to an unscheduled MacBook mother(board) of a meltdown (my emergency backup Linux box won't run TextMate). All told I'm happy with vim's efficiency and power, but I'm mortified at how hard it is to get the kind of word wrap that even stupid HTML textareas achieve with no apparent effort.  Consider the text  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nullam.  By default, with an 80-character width vim displays this as  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nu llam.  This wrapping doesn't care about whitespace, so sometimes it just slices words into pieces (nullam in this case). Of course, you can turn on word wrap, and get this:  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nullam.  The problem is that vim inserts a newline at the linebreak, which I most emphatically do not want. In other words, I want the text to display exactly as vim displays it with word wrap turned on, but without inserting a newline. (This way it can be pasted into HTML textareas and email programs, among other places.)  Web searches have yielded nothing of use, despite diligent effort; I hope StackOverflow can succeed where my Google-fu has failed. How to split one string into multiple strings separated by at least one space in bash shell? Let $(a_n)$ be a sequence of positive terms and suppose that $\sum_\limits{n=0}^{\infty} a_n$ converges. Show that $\sum_\limits{n=0}^{\infty} a_n^2$ converges.   This is in the section on the Comparison Test so that must be what I'm supposed to use.  But I don't see how.  $(a_n)^2$ might be smaller or larger than $a_n$ depending on $a_n$.  And I can't use the Comparison Test with some other series because there's no info here about how fast $\sum a_n$ converges.  Hmm.  Any hints? I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it? Hide a Subdomain from Google/Search Engine SEO Result? In the , the key $k = (a, b)$ should have the following property, i.e. GCD of $a$ and the modulus should be 1:     $\gcd (a, m) = 1$, where $m$ is the set of possible moduli   My question is, why it is required that $a$ should have the above property? What is the significance of this? And why is $b$ excluded from this condition? There is a way to use my Galaxy tab s like a wacom table, on linux? I read some articles, and seen some videos about a github project that do a sort of thing, but it's a little bit outdated things.I wanna see if this thing is possible to do, nowadays. Thanks I am trying to help a friend who is very concerned that she has just lost all of the data on her 32GB SanDisk microSD card.  I told her I would post this and see if anyone could offer help.  Here are the details:   The SanDisk microSD card is less than two years old.  Probably only about a year old.  The store will take it back, but that doesn't really help.  She wants her data. She used the card in her Samsung Galaxy S4 smartphone for over a month.  The phone was never exposed to water or temperature extremes. Today, her phone suddenly said something like "Your SD card is blank or an unsupported format". Her phone was then unable to read the SanDisk microSD card at all.   Here's what she has tried:   Rebooting her phone.  Had no effect. Taking out the SanDisk microSD card and putting it inside a full-size SD card adapter, and inserting it into a Windows 7 SP1 computer.  The computer did not recognize its presence at all. Putting the SanDisk microSD card into a USB 2.0 external card reader and inserting the card reader into a USB port on a Windows 7 SP1 computer.  The computer recognized the card reader and installed drivers for it.  The card reader shows up as 'Removable Disk' in Windows Explorer.  Clicking on 'Removable Disk' in Windows Explorer results in the error "Insert disk: Please insert a disk into the Removable Disk". After trying #3, tried opening the 'Removable Disk' in FreeCommander.  Doing so results in the error "The device is not ready".   She does not have access to a linux box, but does have access to her Android phone.  What can she do to recover the data on her 32GB SanDisk microSD card? I'm using a custom class file.  My main file looks like this:  \documentclass[12pt,a4paper]{thesis} \begin{document} \frontmatter     \pagenumbering{alph}      \pagenumbering{roman}     \clearpage      \cleardoublepage     \tableofcontents     \clearpage     \listoffigures     \clearpage     \listoftables   \mainmatter        \part{test}       \input{test} \end{document}   If I put this in my test.tex it does work:  \chapter{test} \label{chapter:test} test    When it becomes more text (lets say 40x the word "test") I get the error stated in my question title.  To test it I brought my class file down to this:  \ProvidesClass{thesis} \NeedsTeXFormat{LaTeX2e}  % Based on the memoir class \DeclareOption*{\PassOptionsToClass{\CurrentOption}{memoir}} \ProcessOptions \LoadClass{memoir}   I'm totally confused what is causing my error now.  This is the complete error:  ! pdfTeX error (font expansion): auto expansion is only possible with scalable  fonts. \AtBegShi@Output ...ipout \box \AtBeginShipoutBox                                                    \fi \fi How can I install Ubuntu encrypted with LUKS with dual-boot? The Ubuntu installation disk has an option to install Ubuntu encrypted using LUKS. However, there is no option to perform an encrypted installation along-side existing partitions for a dual-boot scenario.  How can I install Ubuntu encrypted alongside another partition from the live disk?