I am trying to figure out how I can prove that there exits $a\in R$ such that $a^3=2$ using the completeness axiom ?  I know that the completeness axiom for the real numbers states that for every non empty set $S$ which is bounded above, then the supremum of $S$ exists.So in the above example, would the supremum be equal to $2$ ?  How do I go about proving the above using the completeness axiom ? Could someone provide a proof for this problem ?
I am trying to prove the existence of the square root of $2$.  I have some steps with a very vague explanation and I would like to clarify.  The proof: Let  $$S=\{x\in\mathbb R\mid x\geqslant 0 \text{ and } x^2&lt;2\}.$$  I understand the proof of LUB, ∝   and so I am at the step where $\alpha^2=2$.  I know that we are to prove by contradiction so we state let $\alpha^2 &lt;2$ and $\alpha^2 &gt;2$.  Now my instructor wants us to use the Archimedean Axiom $1/n = \varepsilon$.    $(\alpha^2 + 1/n)^2$  then what.....
How to open a web page from my application? What causes javac to issue the "uses unchecked or unsafe operations" warning How can I determine how much bleed to use? Version tags reloaded I'm getting sick of version numbers in tag names (aka version tags). Look,      I just want them all to be just . They are all about JSF. I browse and answer questions about JSF. I don't want to browse in all those different tags. I don't want gold badges for both  and . I can't hammer a duplicate  question because I don't have a gold badge for that. Across years I observed that a version tag is usually only picked by the asker when a &quot;relatively new&quot; or &quot;relatively old&quot; version is being treated in the question. When the &quot;currently mainstream&quot; version is being treated,  or when the asker is &quot;more proficient&quot; in the language, then askers often just pick the versionless . While moving in time, this may cause confusion by future readers when the version isn't explicitly specified anywhere in the question. I try to maintain them by adding/removing the version tag depending on its relevance, and by adding the versionless tag when it's absent. It only feels a neverending story. I'm sure other tag caretakers also have mixed feelings on this. I propose to normalize the version from the tag name:  Ban version numbers in tag names in the model (not in the UI). Add an optional single-choice version field to the Tag entity in the model. Users with edit privileges (no suggested edits please) must be able to change versions. Users with gold tagbadges must be able to add and edit versions. I'd imagine the version edit thing to be reachable via that tag info popover (when you hover the tag). Tags themselves however should look exactly the same, the version just separated with e.g. a hyphen from the tag name.  When in place, migration of existing version tags should be piece of cake. There's apparently already a script in place which recognizes version tags, it could be reused to re-match current tags and extract the version from it into the new version field. The alternative would be to automatically add the versionless tag when a version tag is being used. But this is IMO only tag pollution.  I'm posting this on meta.se instead of meta.so because version numbers aren't only on SO a problem, but on basically all HW/SW related sites. Inheriting constructors no space left in '/' but a lot space left in '/home' i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' Cannot edit posts on iOS app When I try to submit an edit in the iOS app it returns this message:        Text:     Account is not allowed to suggest edits   What happened and can it be fixed please? Is it possible to flash a smartphone like the Samsung Galaxy SIII or an LG Optimus 4X with a vanilla Android ROM like the one that has been flashed on Galaxy Nexus or Motorola phones (I think)?  I am asking because manufacturers like Samsung, LG, HTC tend to make ROMs for their smartphones and tablets with preinstalled launchers like Touchwiz or HTC Sense. They explain their decision by saying that they want to make their devices to look unique against their rival devices. The problem is that Android updates for their phones and tablets get delayed and there are incidents in which some devices - which are more than capable of running the upcoming version of Android - fall way behind or never get updated at all.   So, what I am basically asking here is if there is a way to install vanilla Android, excluding CyanogenMod or any other modified vanilla ROMs, on an Android based smartphone? Evaluate  $$\int\frac{e^xdx}{(e^{2x}-1)(e^x+1)}$$  I tried to decompose the integrand into A and B partial fractions.  No success. What can I do to prevent passport stamps being put on blank pages? I'm running low on blank passport pages which are sometimes needed for full page visas such as those issued by Armenia, India, and Vietnam.  When I crossed the border into Turkey the other day I got my visa-on-arrival and stamp, but instead of putting it on a partly used page, which I have many, the guy at customs and immigration put them on a previously blank page. Now I have one fewer!  What can I do to prevent or at least reduce the chances of this happening?  It's worth remembering that there's likely to be a language barrier and it's always best to be ultra-nice to these border officials.  Has anybody heard of any trick such as sticking bit of paper in your passport, either blank, or looking a bit like visas? I have noticed that nobody ever stamps a page that has a remnant of something once stapled in there in Japan. There is a kind of stub that had its major part torn off along a perforation. Might I be able to emulate this somehow or would that even be worse making the blank pages invalid for full page visas due to having holes from staples? I know some things can get very picky when it comes to customs and immigration. Why isn't the symmetric twin paradox a paradox? Two twin sisters synchronize their watches and simultaneously (from the earth frame) depart earth in different directions. Following a predetermined flight plan, each sister accelerates identically to 99.9%c and then returns home at the same time (again in the earth frame). The observers on earth see each twin as having aged identically, as the symmetry of the problem dictates. Each twin however should see the other as having aged... What gives? How do I render a Twig template and load a CSS file from a controller? Is this considered a good practice for rendering a Twig template and loading a CSS file from a controller?   DefaultController.php  namespace Drupal\acme\Controller;  use Drupal\Core\Controller\ControllerBase; use Drupal\Core\DependencyInjection\ContainerInjectionInterface; use Symfony\Component\DependencyInjection\ContainerInterface; use Drupal\Core\Template\TwigEnvironment;  class DefaultController extends ControllerBase implements ContainerInjectionInterface  {   /**   * @var Drupal\Core\Template\TwigEnvironment   */   protected $twig;    public function __construct(TwigEnvironment $twig)   {     $this-&gt;twig = $twig;   }    public static function create(ContainerInterface $container)   {     return new static(       $container-&gt;get('twig')     );   }    /**    * hello    * @param  string $name    * @return string    */   public function hello($name)    {      $template = $this-&gt;twig-&gt;loadTemplate(       drupal_get_path('module', 'acme') . '/templates/hello.html.twig'     );      $markup = [       '#markup' =&gt; $template-&gt;render([ 'name' =&gt; $name ]),       '#attached' =&gt; [ 'css' =&gt; [           drupal_get_path('module', 'acme') . '/assets/css/acme.css'         ]       ]     ];      return drupal_render($markup);   } }   acme.routing.yml  acme_hello:   path: '/acme/hello/{name}'   defaults:     _content: '\Drupal\acme\Controller\DefaultController::hello'     _title: 'acme Title'   requirements:     _permission: 'access content'   hello.html.twig  &lt;div class="acme-hello-text"&gt;   &lt;h1&gt;Hello {{ name }}!&lt;/h1&gt; &lt;/div&gt; "You just can't" vs. "you can't just " I'm a bit confused about this. Which expression is correct?     You can't just do that.    or      You just can't do that.   I'm trying to say:     You can't just bash an ideology because of what someone has said/done. This is a  about CNAMEs at the apices (or roots) of zones   It's relatively common knowledge that CNAME records at the apex of a domain are a taboo practice.  Example: example.com. IN CNAME ithurts.example.net.  In a best case scenario nameserver software might refuse to load the configuration, and in the worst case it might accept this configuration and invalidate the configuration for example.com.  Recently I had a webhosting company pass instructions to a business unit that we needed to CNAME the apex of our domain to a new record. Knowing that this would be a suicide config when fed to BIND, I advised them that we would not be able to comply and that this was bunk advice in general. The webhosting company took the stance that it is not outright forbidden by standard defining RFCs and that their software supports it. If we could not CNAME the apex, their advice was to have no apex record at all and they would not provide a redirecting webserver. ...What?  Most of us know that  insists that A CNAME record is not allowed to coexist with any other data., but let's be honest with ourselves here, that RFC is only Informational. The closest I know to verbiage that forbids the practice is from :     If a CNAME RR is present at a node, no other data should be   present; this ensures that the data for a canonical name and its aliases   cannot be different.   Unfortunately I've been in the industry long enough to know that "should not" is not the same as "must not", and that's enough rope for most software designers to hang themselves with. Knowing that anything short of a concise link to a slam dunk would be a waste of my time, I ended up letting the company get away with a scolding for recommending configurations that could break commonly used software without proper disclosure.  This brings us to the Q&amp;A. For once I'd like us to get really technical about the insanity of apex CNAMEs, and not skirt around the issue like we usually do when someone posts on the subject.  is off limits, as are any other Informational RFC applicable here that I didn't think of. Let's shut this baby down. Is the "live" in "He only lived a few days after the accident" intrasitive or transitive? An example sentence from the :  [I] He only lived a few days after the accident.  [I] means &quot;intransitive verb&quot;. He (subject) + lived (verb) + a few days (noun). What part of speech is a few days? From the :  number 5  way of life[intransitive always + adverb/preposition, transitive] to have a particular type of life, or live in a particular way   used 'transitive verb' like 'She lives a very busy life.' From the ELU question :  She is 16 year old.  '16 years' as acting as an adverb. My questions are:  In &quot;He lived a few days.&quot;, is a few days an object (noun) or an adverb? How is it different from &quot;He lived for a few days.&quot;? &quot;I walked ten miles.&quot; Number + measurement — how can I deal with 'part of speech'? Does shoving a creature break its grapple on another? A situation came up in my session where a fighter used his Shield Master Shove feature to shove a Giant Crocodile that was currently restraining another PC using it's bite+grapple ability.  I did a quick check at the time and did not find anything related to shove, although in the  condition description it states:     The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.   I ruled at the time that a shove would count in a similar way to being "hurled away" however I wasn't sure and said I would find more concrete information for the future.  I have searched high and low for somebody else asking this specific question but have yet to find it. I saw some reference to a Jeremy Crawford tweet about involuntary movement and grappling however I am unable to find it.  If a third party "shoves" a creature that is grappling another, does it only shove the grappler away, or does it move both the grappler and grappled?  I drew a diagram:   (; this question relates to if a third party shoves the grappler instead, so it's not a duplicate.) What is the autocorrelation for a random walk? Seems like it is really high, but this is counterintuitive to me. Can somebody please explain? I am very confused by this issue and would appreciate a detailed, insightful explanation. Thanks a lot in advance! How do I change player name in Minecraft smp in offline mode on a Mac? If I have the following enum:  public enum ReturnValue{     Success = 0,     FailReason1 = 1,     FailReason2 = 2     //Etc... }   Can I avoid casting when I return, like this:  public static int main(string[] args){     return (int)ReturnValue.Success; }   If not, why isn't an enum value treated as an int by default? Add lines to cron from script Is there a way to add lines to a user's cron via script?  I usually do it using crontab -e, but I would like to automate this task with a shell script.