let $A = \{m + n \sqrt{2}\}$ where $m,n$ are integers, then  $a.$ $A$ is dense in $R$.  $b$. $A$ has no limit point in $R$.  $c$. $A$ has only countably many limit points in $R$.  $d$. only irrational numbers can be limit point of $A$  Set $A$ is similar to the set $N \times N$, so all points are isolated in $A$. therefore option $b$ should be correct. In answer key, option $a$ is marked as correct. Please suggest if I am doing something wrong.
I am having trouble proving the statement:     Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon &gt; 0$, the intersection of $S$ and $(0, \epsilon)$ is nonempty.
If $A$ is a non-empty set, then $A \notin A$ Exercise 3.2.2 Use the axiom of regularity and the singleton set axiom to show that if $A$ is a set then $A \notin A$.  Axiom of Regularity: If $A$ is a non-empty set, then there is at least one element x of A which is either not a set, or is disjoint from A.  Singleton Set Axiom: If $a$ is an object, then there exists a set $\{a\}$ whose only element is $a$, we refer to $\{a\}$ as the singleton set whose element is $a$ Furthermore, if $a$ and $b$ are objects, then there exists a set $\{a,b\}$ whose only elements are $a$ and $b$    I think I can give the proof when A is the non-empty set which contains finite elements. However, can anyone give any hint for the general case?  Suppose $A \in A$   Then eixsts $x$ such that either $x$ is not a set or $x \cap A$=$\emptyset$  Then it is clear thar $x \ne A$ (if $x = A$, then $x \cap A \ne \emptyset$)  We observe that as an element, $A$ $\ne$ $x$, $A$ $\in$ $A$  Then $A$ $\in$ $A-x$   Define the new set $A-x$ as $A_1$  then exists $x_1$ such that $A \in A_1-x_1$  After finite steps  we find that $A \in \emptyset $  which leads to contradiction. On iOS, while watching a video in YouTube app, when I go to the Home Screen, the video stops playing. I found no setting to enable it.  I know it is available for Android, but is there a way to do so in iOS? 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The thing is, I'm modelling something that's not going to be an animation, but a static object (something like a statue/figurine). So I don't need the animation, only the shape at a particular frame that I find convenient. I'd like eventually once this is done, to also modify that deformed grid.   I've tried to follow what was said , but couldn't figure out what chipmasque was saying. I'm quite new to blender (been using it on and off, just enough to forget everything).  I'm using blender 2.63, if it is of any relevance. My display driver's maximum screen width is 1280 pixels.   I need to test a website with resolution 1920&times;1080 pixels. Or 3000&times;5000.   Different browsers is optional. Prove norm doesn't come from inner product. Please help me prove this. I'm not sure how to apply the parallelogram law to the norm. Please provide a mobile-optimized captcha page Foreign key constraint on array member? Suppose I have a table containing job roles:  CREATE TABLE roles (   "role" character varying(80) NOT NULL,   CONSTRAINT "role" PRIMARY KEY (role) );   Suppose I further have a table, users, and each row (a specific user) can have an arbitrary number of job roles:  CREATE TABLE users (   username character varying(12) NOT NULL,   roles character varying(80)[] NOT NULL,   CONSTRAINT username PRIMARY KEY (username) );   I should probably make sure that each member of users.roles[] exists in roles.role.  It seems to me that what I want is a foreign key constraint on each member of users.roles[] such that if references roles.role.  This doesn't seem possible with postgres.  Am I looking at this the wrong way?  What is the suggested "right" way to handle this? I typed up an assignment with a lot of numbers in LaTeX and turned it in to my adviser.  When he returned it to me with some corrections, one of which was a lot of $ around my numbers.  I understand that $ are necessary around some mathematical formulas, but I was just using numbers like -2 and 4.  I did not see any change in the font of the numbers when I added the $ either.  Since my adviser did not want to give me an explanation, what is the rationale? Sitemaps - one per subdomain or one for the base domain? I have more than 5 subdomains on a project, say developer, www, staff and some more, what is the best practice in creating sitemaps?   All subdomains that share a same sitemap at base domain (example.com/sitemap.xml) ? or different sitemaps for different subdomains?(developer.example.com/sitemap.xml and www.example.com/sitemap.xml)   Since the subdomains are several parts of the main application, what are the pros and cons of each, and what should I consider while choosing the one over the other? 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In older versions of Windows and even in Windows 10 previews you could enable the built-in guest account in control panel     Control Panel\User Accounts\User Accounts\Manage Accounts     In Windows 10, the Guest entry is missing.  Using Computer Management or net user you can still enable the guest account, but even then it doesn't show up on the logon screen.  In the Local Security Policy secpol.msc:     Local Policies - User Rights Assignment - Deny log on locally   we can see guest listed, this wasn't the case in older versions of Windows.  After removing guest from that policy, it is listed on the logon screen and we can using it to log on.  However, the whole Explorer shell is crashing and restarting constantly. The desktop is blinking and you can't click anything.  So the guest account is totally broken.   There is a  about this showing that some code was quickly added to the control panel to not show the guest account. So Microsoft does know about the problem.  While I don't really expect anybody to know what they've done to break this, at least this explains why the guest account is currently not listed. What are the alternatives to the FHS? I'm a long time Linux user for over 15 years but one thing I hate with a passion is the mandated directory structure. I don't like that /usr/bin is the dumping ground for binaries or libs in /usr/lib, /usr/lib32, /usr/libx32, /lib, /lib32 etc... Random stuff in /usr/share etc. It's dumb and confusing. But some like it and tastes differ.  I want a directory structure where each package is isolated. Imagine instead if the media player dragon had it's own structure:  /software/dragon /software/dragon/bin/x86/dragon /software/dragon/doc/README /software/dragon/doc/copyright /software/dragon/lib/x86/libdragon.so   Or:  /software/zlib/include/zlib.h /software/zlib/lib/1.2.8/x86/libz.so /software/zlib/lib/1.2.8/x64/libz.so /software/zlib/doc/examples/... /software/zlib/man/...   You get the point. What are my options? Is there any Linux distro that uses something like my scheme? Can some distro be modified to work like I want it (Gentoo??) or do I need LFS? Is there any prior art in this area? Like publications on if the scheme is feasible or unfeasible?  Not looking for OS X. :) But OS X-inspired is totally ok.  Edit: I have no idea how PATH, LD_LIBRARY_PATH and other environment variables that depend on a small set of paths should work out. I'm thinking that if I have the KDE editor  installed in /software/kate/bin/x86/bin/kate then I'm ok with having to type the full path to the binary to start it. How it should work for dynamic libraries and dlopen calls, I don't know but it can't be an unsolvable engineering problem. Repeating (or not) "to" in a sentence involving infinitives set in parallel I googled it but I couldn't find the answer... maybe I didn't seek it in the proper way. My question is if &quot;to&quot; must be repeated in sentences such as the following ones:   In the past, women used to do the housework and to take care of children.  The aim of this meeting is to control their progress and to discuss the new working plan.    If I had to bet, I would omit the second to... But I am not sure about it. How to use an existing database with an Android application UNITY : I want to make my UI text fade in after 5 seconds I know this question has been asked and asked but none of the topics i visited gave me an answer. I'm making a game programmed in c# where you control a boat and i want my UI to appear smoothly by fading in after X seconds. May someone help me? Subfields of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ Suppose one doesn't know any Galois theory, but they are familiar with the basic theory of field extensions. Then how would one justify the following picture?    I'm pretty sure they're just saying these are the known subfields, and there could be more but they just write down those ones (which seems weird, which is why I'm asking). Or have they deduced that this is a comprehensive list of subfields just by using basic theory? This occurs on page 573 of Dummit and Foote's Abstract Algebra. Why is $\frac{\operatorname dy'}{\operatorname dy}$ zero, since $y'$ depends on $y$? I know that $\frac{dy'(x)}{dy}=0$ (where $y'=\frac{dy(x)}{dx}$). The reason explained is that $y'$  does not depend explicitly on $y$. But intuitively, $y'$ depends on $y$, since if you vary $y$ you will modify $y'$. Why is my reasoning wrong (my reasoning sounds like it's related to functional calculus, instead of standard calculus)?  I tried to write  $\frac{dy'(x)}{dy}=\frac{d}{dy}\frac{dy}{dx}=\frac{d}{dx}\frac{dy}{dy}=\frac{d}{dx}1=0$, but this proof doesn't convince me.  I think that other way to see this is saying that if a function is of the form $f(y)$, it will not dependend on the variable $y'$.  But the same way you write $f(y)=y^2$, you could write $f(y)=\frac{d}{dx}y$, which clearly depends on $y'$. So I don't know if there are some types of operations which are restricted (for example, taking limits):  Note: The problem raises in the context of Classical Mechanics, where: $\frac{\partial L(\dot{x},t)}{\partial x}=0$. How to change this matrix with respect to standard basis? Given the basis $\beta = \{ (1, 1, 0),\ (1, 0, -1),\ (2, 1, 0)\}$ and the matrix:  $$ A =         \begin{bmatrix}         1 &amp; 1 &amp; 0 \\         1 &amp; 1 &amp; 0 \\         0 &amp; 0 &amp; 1 \\         \end{bmatrix} $$  with respect to  $\beta$.  How to change this matrix with respect to standard basis?