Proving that $\lim\limits_{x\to\infty} xf(x) = 0$ given certain conditions
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
I know I've seen somewhere how to typeset 1/2 as ½, but with arbitrary fractions (that is, a macro that takes two arguments and makes the first denominator and the second numerator...). Now I can't seem to find it.  How do I typeset an arbitrary fraction the same way as ½? Infinite partition of $\mathbb N$ by infinite subsets Is my expansion of the state $| x \rangle$ correct? What to do when the correct answer to a question changes over time? My question concerns . I asked the question, and at the time selected the correct answer. However, time passes and libraries get updated, and now a different answer is actually correct.   I should probably select this new answer as the correct answer, but I feel bad for the person that will lose 15 reputation points even though is answer was the a good answer at that given point in time.  Any suggestions? In my profile I saw this:     Accounts (48) - 1 hidden account   as seen in this image:        So, what is this "hidden account"? And how do I see it? Is there a canonical explanation for how Ant-Man's weight works when he shrinks down that's consistent with the various events depicted in the film?    I think the explanation given in the film is roughly "the suit compresses the space in-between atoms, so your size decreases, your density increases, and your total mass stays the same", however it seems to be portrayed in contradictory ways.  For instance:   He can float in water, and even be sent shooting through the air by a relatively gentle flow of water filling a bathtub.  This wouldn't happen if he retained his original mass, as he'd essentially be the densest thing on the planet and should sink like a stone even in a vat of molten lead. Upon landing after being ejected from the bathtub, the ceramic tile underneath him shatters.  That takes a significant amount of force, and similar effects are used at other points in the movie where he falls onto (or straight through) various things. He's able to get sucked into a vacuum-cleaner.  One that didn't even appear to be a high-end, extra powerful vacuum-cleaner, let alone one capable of sucking up a 200 pound piece of dust.   He can run along the barrel of a gun as someone shoots it, apparently without causing them to drop the weapon or even have trouble continuing to aim it.     Does a more consistent explanation exist in other media, that would fit with the above examples?  Or is this simply a case of the power being made to do whatever happened to be most convenient at the time? How to put a unique material on duplicated objects separately? I've a sphere and then duplicated it. In object tree they are displayed as three siblings. I selected on of them and created a green glass material through node editor however this material was assigned to all three spheres. I confirmed that my selection is only on one sphere. What am I missing? Windows 8 upgrade Recovery Disk I have upgraded to Windows 8 from Win 7. I want to create a recovery disk for Windows 8, just in case something goes wrong. I didn't find the option of creating it from the control panel.  How do I create a Recovery CD/DVD? Alarm is working even when the phone is powered off I just discovered that if I set the alarm clock and then power off the phone (by holding the power button and then selecting "Power Off") the alarm will still go off when the time comes.  Is this expected behavior? It makes me feel like "power off" doesn't actually power off the phone.  I have a Huawei Y530-U051 running android 4.3. How to calculate such sums of Legendre symbols? Soving a Complex Integral along a circle How to convert .bib generated by Mendeley for use with amsrefs I'm using TeX live 2010.  I have a .bib file generated from Mendeley, and  I want to use that with amsrefs, but it's confusing and the amsrefs documentation is somewhat hard to follow: how do I massage the .bib file into something that amsrefs can stomach. (I'm using amsart as the document class). Alice has a secp256k1 key pair that she used some time ago when she communicated with Bob. Bob can no longer user secp256k1 but he only uses ed25519 instead.  Is there a secure algorithm that Bob can use to derive an ed25519 pubkey from original Alice's secp256k1 pub key and also Alice can derive the matching ed25519 private key from her original secp256k1 private key such that they can communicate securely via ed25519 encrypted messages?  Thanks! I need to convert some files to PDF and then attach them to an email.  I'm using Pear Mail for the email side of it and that's fine (mostly--still working out some issues) but as part of this I need to create temporary files.  Now I could use the  function but it sounds like it creates a file on the filesystem, which isn't what I want.  I just want a name in the temporary file system (using ) that won't clash with someone else running the same script of the same user invoking the script more than once.  Suggestions? grub2 not able to find the new kernel This is very strange issue, i have CentOS7 running on HP server and everything was working till date until i did "yum install kernel" after reboot it drop me in grub&gt; shell that means it won't able to read or find grub.conf file.   in grub&gt; i found grub.conf has still old files there is no new entries of new kernel   I did this, and found there is no new kernel entries but if i boot with old kernel then i can see latest grub.conf (with new kernel)  grub&gt;cat /grub2/grub.conf   Question: where grub2 getting that old file from? why it's not able to find new kernel ? How to upgrade from 18.10 to 19.10, using the command line? I'm trying to upgrade from 18.10 to 19.10, using sudo do-release-upgrade however I only get an error: An upgrade from 'cosmic' to 'eoan' is not supported with this tool. I performed the steps to update my sources.list to point to old-release server, as suggested here:  This did not help. I get the same error. Zero probability and impossibility I read a comment under :     There are plenty of events that can occur that have zero probability.   This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering    if zero probability and impossibility mean the same? if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?   Thanks and regards! I work in D8, I create a custom module first_module. By modules create a url and display all nodes by *.html.twig file.  Write code in controller.php  public function contentthree() {    $connection = Database::getConnection();    $query = $connection-&gt;select('node', 'n');   $query-&gt;fields('n', ['nid', 'type']);    $groups = $query-&gt;execute()-&gt;fetchAll();    foreach ($groups as $group) {     $events[] = [$group-&gt;nid, $group-&gt;type];   }    return [     '#theme'  =&gt; 'first_module_display',     '#events' =&gt; $events,   ]; }   In *.module  function first_module_theme($existing, $type, $theme, $path) {    return [     'first_module_display' =&gt; [       'variables' =&gt; ['events' =&gt; NULL],       'template'  =&gt; 'first-module-list',     ],   ]; }  function _get_node_details() {    $output = 'Welcome D8';    $output .= 'Node Details';    return $output; }   and create templates first-module-list.html.twig and display data  {%for key, data in events %}  {%for key1, data1 in data %}     {{ data1 }}  {% endfor %}  {% endfor %}   work fine and showing data correct. But, my question is how to call function _get_node_details() and how to render node details in first-module-list.html.twig files? This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? Word for seeing both sides of an argument