How to prove that:  $|x+y| = |x| + |y|$ if and only if $xy \ge0$    and  $|x-y| = |x | - |y|$ if and only if $xy \le0 $  and $|x| \ge |y|$?  I understand that these have similar approach and that each needs two proofs (that A implies B and that B implies A), but for know I've got as far as just writing $xy \ge 0$ and making different operations with it.
How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the three conditions happens.
Bravo for allowing comment editing, but what's the explanation for enforcing a five-minute window?  I think this might be over-engineering. If I have a spelling mistake or an inaccuracy I'd like to fix, I don't think the fact that five minutes has elapsed should prevent me from fixing it.  Further, if you are worried people will edit their comments in a way that makes the comment thread nonsensical, they can already do this by deleting their comment. So, limiting the ability to edit isn't really helping much in that regard.  Update  If people don't like this idea, I'm definitely open to hearing your reasons. For now, it seems like an unnecessary limitation of a quite awesome feature.  Update 2  Regarding John's scenario...     Me: Obama is cool!   You: You are right, man!      My edit: Obama is an idiot!   You: You are right, man!   ...this problem can still occur with deletions...     user1: Obama is cool!   user2: No, McCain is cool!   you: You are right, man!        user1: Obama is cool!   ---- deleted -----   you: You are right, man!     So, yes, editing could be abused as you suggest, but discontinuities can occur already because of deletions (and as someone else mentioned, because upvoted comments get prioritized upward).  As for Jeff's argument about rate limiting, I agree that is important, but I don't understand why you can't just use a CAPTCHA to prevent too many edits in a short period of time. If a function has a finite limit at infinity, does that imply its derivative goes to zero? I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L &lt; \infty$. Then must it be the case that $\lim\limits_{x\to +\infty}f'(x) = 0$?  It looks like it's true, but I haven't managed to work out a proof. I came up with this, but it's pretty sketchy:  $$ \begin{align} \lim_{x \to +\infty} f'(x) &amp;= \lim_{x \to +\infty} \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\ &amp;= \lim_{h \to 0} \lim_{x \to +\infty} \frac{f(x+h)-f(x)}{h} \\ &amp;= \lim_{h \to 0} \frac1{h} \lim_{x \to +\infty}[f(x+h)-f(x)] \\ &amp;= \lim_{h \to 0} \frac1{h}(L-L) \\ &amp;= \lim_{h \to 0} \frac{0}{h} \\ &amp;= 0 \end{align} $$  In particular, I don't think I can swap the order of the limits just like that. Is this correct, and if it isn't, how can we prove the statement? I know there is  already, but I think this is different in two aspects. First, that question assumes that $\lim\limits_{x \to +\infty}f'(x)$ exists, which I don't. Second, I also wanted to know if interchanging limits is a valid operation in this case. I want to append several shapefiles in QGIS. The ftools 0.6.1 add-on is installed.   But under Vektor>Datenmanagement-Werkzeuge>Shapedateien zu einer zusammenführen (=append shapefiles) is only the possiblility to choose the Eingabedatei (input file) and the Ausgabedatei (output file) but not a further Datei (file) for the append itself? (Even though there are  several shapefiles in the same folder)  //Furthermore the icon "In neue Projektion exportieren" (export to new projection) is also not appearing under the: Vektor>Datenmanagement-Werkzeuge? Do I need a further add-on?//  Has anyone an idea why I can't choose a shapefile for the appending?  Thanks in advance! phil Why would a single celled organism evolve to be multi-celled? I read a story this week on Richard Lenski who has been 'evolving' E. coli for more than 50,000 generations now.  One comment I read was from someone who doesn't accept Evolution who pointed out that we haven't seen a single celled organism 'evolve' into a multi-celled organism.  Another person responded and said that a bacteria is not going to evolve into something that isn't a bacteria.  So, if Evolution created single celled organisms and then multi-celled organisms how might that change have happened?  And is it possible to recreate that set of driving forces to make a bacteria something other than a bacteria?  To that end, what advantage does being multi-cellular have over being unicellular (if that's even a word)? How reliable is validating email domain by reverse DNS MX record checking An English book tells me that the following sentence is correct:     This publication is made from paper from sustainable forests.   Shouldn't it be of paper instead? How does polynomial long division work? How do I clear command prompts in minecraft? Join tables from two different server Show if a user is online, idle, or offline on their account page? I want to display the status of the user on the account page. I added the following code in the theme I am using. It works, but users are never displayed offline. Why? What is wrong in the code? user.html.twig &lt;div class=&quot;bs-field-status&quot;&gt;   {% if status == 'Online' %}     &lt;i class=&quot;user-online fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Online   {% elseif status == 'Absent' %}     &lt;i class=&quot;user-absent fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Absent   {% else %}     &lt;i class=&quot;user-offline fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Offline   {% endif %} &lt;/div&gt;  bootstrap_subtheme_front_office.theme &lt;?php /**  * @file  * Bootstrap sub-theme.  *  * Place your custom PHP code in this file.  */  use Drupal\Core\Database\Database;  /**  * Implements hook_entity_presave().  */ function bootstrap_subtheme_front_office_preprocess_user(&amp;$variables) {   // get user object   $user = $variables['elements']['#user'];   //- The user has logged in at least once   if ($user-&gt;getLastLoginTime()) {     if (account_is_logged_in_less_then_thirty_minutes($user-&gt;id())) {       $status = 'Online';     }     else {       $status = 'Absent';     }   }   else {     $status = 'Offline';   }   $variables['status'] = $status; }  /**  * @param $uid  *  * @return bool  */ function account_is_logged_in_less_then_thirty_minutes($uid) {   $connection = Database::getConnection();   $query = $connection-&gt;select('sessions', 'sessions')     -&gt;fields('sessions', ['sid', 'uid', 'timestamp'])     -&gt;condition('sessions.uid', $uid, '=')     //- chef if the user was online in 30 minutes (60 * 30)     -&gt;condition('sessions.timestamp', \Drupal::time()         -&gt;getRequestTime() - (60 * 30), '&gt;')     -&gt;execute();   //- Get result.   $results = $query-&gt;fetchAll(\PDO::FETCH_OBJ);   return (count($results) &gt; 0) ? TRUE : FALSE; } Atomic clocks were used in Hafele and Keatings experiment which supposedly helped to prove time dilation.    How can we be sure other forces didn't act upon the clocks as a result of their being at different speeds and elevations? Perhaps the elapsed time may have been the same for all and only appeared different as a result of the clock's atoms behaving differently in different environments.  How might the presence of dark matter (presumably all over the place) impacted the clocks if say the dark matter is stationary relative to the earth's rotation?  What experiments have been done to test the constancy of clocks at different speeds and elevations? I use both GNOME and KDE SC. In GNOME, I use Docky but I do not use it with KDE. So, I would like Docky to start up only when I login into GNOME not KDE. In KDE, there is an option to make a program autostart only in KDE. But I cannot find any such option in GNOME.    How can I do this?    Thanks in advance. Short story identification: society partitioned off by high wall How does a lurker gain reputation to receive privilege for commenting? Vertical text (not in table) Forcing a SimpleXML Object to a string, regardless of context Let's say I have some XML like this  &lt;channel&gt;   &lt;item&gt;     &lt;title&gt;This is title 1&lt;/title&gt;   &lt;/item&gt; &lt;/channel&gt;   The code below does what I want in that it outputs the title as a string  $xml = simplexml_load_string($xmlstring); echo $xml-&gt;channel-&gt;item-&gt;title;   Here's my problem.  The code below doesn't treat the title as a string in that context so I end up with a SimpleXML object in the array instead of a string.  $foo = array( $xml-&gt;channel-&gt;item-&gt;title );   I've been working around it like this  $foo = array( sprintf("%s",$xml-&gt;channel-&gt;item-&gt;title) );   but that seems ugly.  What's the best way to force a SimpleXML object to a string, regardless of context? Sometimes writing braces and omitting them produces the same result:  \tilde{\psi}  \mathrm{e}^{\mathrm{i} \pi}   vs  \tilde \psi   \mathrm e^{\mathrm i \pi}   Or (taken from Werner's comment below)   \frac{1}{2} vs \frac12   If that's the case, how bad for the quality of my code is omitting them? Of course, if one compiles a pair of formulas, there is no difference. But, since I ignore how TeX works, my point is: if one writes long codes with these habits,   doesn't TeX go somehow crazy because I'm not writing as it expects?   For instance, does it have any effect on how fast are long documents compiled? Any reason to keep them? Why is an untranslated string not showing up? In my module I have some custom strings. I use these in the t() function. When I go to admin/config/regional/translate/translate and filter the untranslated strings my custom strings do not show here.  What could be going wrong? What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography? Is there a limit to the number of Power Armor sets you can keep? I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep?