Show that $a_n=\sin(n)$ does not converge     My idea:  Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$  So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, $\lim_{n\to\infty} a_{n_l}=-1$  So two inifinite subsequence converge to different limits thus the sequence $a_n$ doesn't converge.   Is that correct ?  Edit:   The contrapositive of the defintion of a limit of a sequence:   $\exists\epsilon&gt;0 : \forall n\in N : \exists n&gt;N \Rightarrow |x_n-L|&gt;\epsilon$  Take $\epsilon =1$ and we know that $\sin(n)$ is bounded so lets take it's supermum: 1  $|1-1|&gt;\epsilon=1\Rightarrow 0&gt;1 \Rightarrow$ Contradiction.
Using only the delta definition of a limit, how can we prove that the sequence $\{a_n\}$, where $a_n = \sin n$, as $n$ tends to infinity does not have a limit?  Thanks!
What is the British English way of referring to a source in-text.  Is it plural because there are two authors, like this:         X and Y (2011) describe ...   Or is it singular because you are referring to a singular source, like this:     X and Y (2011) describes...   So what I am asking is whether you refer to the source or to the authors? Workflow for converting LaTeX into Open Office / MS Word Format Enumerate after \label I have the following code:  \usepackage{hyperref} \newtheorem{thm}{Theorem}  \begin{thm} \label{mythm}    \begin{enumerate}       \item     \end{enumerate} \end{thm}  \ref{mythm}   The ref prints the right number. However, when I click on it, it will lead me to a wrong page (basically, to the beginning of the document).  pdflatex also has some warning.  If I add $ $ between \begin{thm} and \begin{enumerate}, the problem disappears, but now, the enumeration starts a new line.  Is there any way to fix this? How can I securely format a solid-state drive? 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The Node Editor won't work for the object selected I have a Visual Studio 2008 solution with two projects (a Word-Template project and a VB.Net console application for testing).  Both projects reference a database project which opens a connection to an MS-Access 2007 database file and have references to System.Data.OleDb.  In the database project I have a function which retrieves a data table as follows   private class AdminDatabase    ' stores the connection string which is set in the New() method    dim strAdminConnection as string     public sub New()    ...    adminName = dlgopen.FileName    conAdminDB = New OleDbConnection    conAdminDB.ConnectionString = "Data Source='" + adminName + "';" + _        "Provider=Microsoft.ACE.OLEDB.12.0"     ' store the connection string in strAdminConnection    strAdminConnection = conAdminDB.ConnectionString.ToString()    My.Settings.SetUserOverride("AdminConnectionString", strAdminConnection)    ...    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But when I run it from the console application it throws the following exception     ex = {"The 'Microsoft.ACE.OLEDB.12.0'   provider is not registered on the   local machine."}   Both projects have the same reference and the console application did work when I first wrote it (a while ago) but now it has stopped work.  I must be missing something but I don't know what.  Any ideas? Why is the Finder app always open? First time Mac user. I am confused by the fact that the finder application is always open. For example, when switching apps with Cmd+Tab, it is always there, even when there are no windows opened for the app. Cmd-tabbing to Finder with no windows/tabs opened for it has no effect (it seems to me it would more intuitive to open a new tab when this action is chosen).   Furthermore, there is no 'Close app' option when I right click the Finder icon in the dock. Am I misunderstanding something about the Finder app? Is it required for proper functioning of Mac OS? From what I see, Finder is just a file browser, which by no means should be forced to be open 100% of the time. I'm looking for a symbol for skeptical consequence so that I can write that something, B, is a skeptical consequence of something else, A.  In a book I read it's written like \(A \vdash B\) or \(A \sststile{}{} B\) but the horizontal line is wavy like \sim, has the same width as the vertical line and does not vary in width. So, what I'm looking for is a binary relation symbol that is similar to the combination of the first two and the last relation in the following image:    That is, something similar to the following mockup but the wavy line should be such that it has a constant width and the same width as the vertical line:    I've tried  without success. Unsubscribe anonymous method in C# Is it possible to unsubscribe an anonymous method from an event?  If I subscribe to an event like this:  void MyMethod() {     Console.WriteLine("I did it!"); }  MyEvent += MyMethod;   I can un-subscribe like this:  MyEvent -= MyMethod;   But if I subscribe using an anonymous method:  MyEvent += delegate(){Console.WriteLine("I did it!");};   is it possible to unsubscribe this anonymous method?  If so, how? How to apply for US passport card when I already have passport book? I have a US passport book, and I would like to apply for a US passport card (to have in addition to my passport book).  If I apply using form DS-82, submitting my current passport book as proof of citizenship, does it get returned to me still valid, or does it get cancelled?  Is it possible to apply for a passport card using form DS-11 and avoid sending in my current valid passport book (submitting a birth certificate instead)?  I would prefer answers with official sources, which I have not been able to find myself on the US State Dept website or on the forms themselves (which are written quite ambiguously). Managing constructors with many parameters in Java In some of our projects, there's an class hierarchy that adds more parameters as it goes down the chain. At the bottom, some of the classes can have up to 30 parameters, 28 of which are just being passed into the super constructor.  I'll acknowledge that using automated DI through something like Guice would be nice, but because of some technical reasons, these specific projects are constrained to Java.   A convention of organizing the arguments alphabetically by type doesn't work because if a type is refactored (the Circle you were passing in for argument 2 is now a Shape) it can suddenly be out of order.  This question might be to specific and fraught with "If that's your problem, you're doing it wrong at a design level" criticisms, but I'm just looking for any viewpoints. Any Set with Associativity, Left Identity, Left Inverse is a Group I'm trying to give the best experience to users of a very huge system (like a CRM). It has a lot of forms and much of these have a lot of fields. My doubt is about if I should use a single column form (which raises too much vertical scrolling), or a two columns form (which needs less vertical scrolling and uses more horizontal white space) and a last one with four columns (which has few scrolling and uses almost whole white space to try to keep most of the fields above the fold).  It's important to say that our clients hate scrolling, and they always ask why we don't use the horizontal white space when we present a one column form).  So I'm little bit confused about it, because a lot of UX studies and good practices recommend to do not use/avoid multicolumn forms but the clients most of times ask multi column forms.  Our clients are very used to desktop programs and not everybody is very familiar with web based systems.  Below follows some possibilities I would like to share to illustrate an example of a complex form in the system with the three possibilities I mentioned:    In the example above I'm using the one column form (most recommended in UX studies), but with a very long vertical scroll.    In the second example I tried to reduce vertical space by using two columns.    This is the client's favorite one, 4 columns using white space and trying to keep everything above the fold.   The lengths of fields are based on the average of characters amount for each field.   I would like to know your opinion about the best approach to this case. How can I approach implementing the following in Drupal 7?  What I need to do is create a module which defines a new fieldable entity, called 'Company'. I have a list of, let's say, 20 fields that need to be filled out by each Company instance. These questions are predefined and some may contain custom validation.  At the moment, I'm at the point when I can add new fields to the Company entity. This works fine at the moment. My issue is that I need all these fields to be there as soon the module is installed, so adding them via the interface isn't an option.  I was wondering how can I approach this? I'd assume it comes down being able to do what can be done using the 'Manage Fields' UI programmatically. How can I create a Chatter File via Apex? From the Chatter tab I can go to 'Files' and then upload a file without attaching it to anything.   I am wanting to know if there is a way to achieve the exact same thing via Apex? Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance. Today, comparing processors is obviously not just looking at the frequency and the number of cores.  When choosing a new PC, how do you compare processors? Is there some resource, like a site or something that does (not necessarily super-precise) processor comparisons?  Thanks. What does the leading semicolon in JavaScript libraries do?