Prove that if $x_n$ goes to x, then $(x_1x_2\ldots x_n)^{\frac{1}{n}}$ goes to $x$.
This is not a duplicate of this .  The linked question says that it suffices to show that if $(x_n)\to x$ then $(\frac{x_1+\cdots+x_n}{n})\to x$ to prove my question, but how so? I tried using the same strategy as how one proves that if $(x_n)\to x$ then $(\frac{x_1+\cdots+x_n}{n})\to x$, by "splitting" the product in th $N$th term:  $$\sqrt[n]{x_1x_2\cdots x_n}=\sqrt[n]{x_1x_2\cdots x_Nx_{N+1}\cdots x_n}=\sqrt[n]{x_1x_2\cdots x_N} \sqrt[n]{x_{N+1}\cdots x_n}$$  but it seems I can't use for now the definition of convergence of $(x_n)$ because of the $n$th root. I also tried to use a result: if $(x_n)\to x$ then $(\frac{x_n}{n})\to 1$ but don't know if this is true. Sadly I've had no real progress. Any help will be greatly appreciated, thanks in advance!
Potential difference across a zero resistance wire So I started off with electrostatics and everything seemed nice and mathematical and justified and then "DC circuits" happened!  I just cannot understand the model of electron flow in electrical circuits.  Here are my specific doubts-:  1) If potential difference across a tiny cross section of conducting wire is zero, then why on earth does electron flow across that cross section at all? Never mind potential difference across the whole circuit.   2) Is there a constant electric field across a wire connected to a battery? If yes then how is potential difference across a zero resistance wire constant?   Shouldnt it be increasing?  Doesnt it violate ohms law?  If no,  then why do electrons flow at all?  Please take time to consider these doubts and relieve me of my frustration.  I havs searched through the net for this but every answer seems like beating around the bush.  All of the 4 books I have consulted do not address these facts to my satisfaction.   Frankly I think nobody understands this. Working with IPv6 Addresses in PHP After searching around somewhat thoroughly, I noticed a slight lack of functions in PHP for handling . For my own personal satisfaction I created a few functions to help the transition.  The IPv6ToLong() function is a temporary solution to that brought up here: . It will split the IP in to two integers and return them in an array.  /**  * Convert an IPv4 address to IPv6  *  * @param string IP Address in dot notation (192.168.1.100)  * @return string IPv6 formatted address or false if invalid input  */ function IPv4To6($Ip) {     static $Mask = '::ffff:'; // This tells IPv6 it has an IPv4 address     $IPv6 = (strpos($Ip, '::') === 0);     $IPv4 = (strpos($Ip, '.') &gt; 0);      if (!$IPv4 &amp;&amp; !$IPv6) return false;     if ($IPv6 &amp;&amp; $IPv4) $Ip = substr($Ip, strrpos($Ip, ':')+1); // Strip IPv4 Compatibility notation     elseif (!$IPv4) return $Ip; // Seems to be IPv6 already?     $Ip = array_pad(explode('.', $Ip), 4, 0);     if (count($Ip) &gt; 4) return false;     for ($i = 0; $i &lt; 4; $i++) if ($Ip[$i] &gt; 255) return false;      $Part7 = base_convert(($Ip[0] * 256) + $Ip[1], 10, 16);     $Part8 = base_convert(($Ip[2] * 256) + $Ip[3], 10, 16);     return $Mask.$Part7.':'.$Part8; }  /**  * Replace '::' with appropriate number of ':0'  */ function ExpandIPv6Notation($Ip) {     if (strpos($Ip, '::') !== false)         $Ip = str_replace('::', str_repeat(':0', 8 - substr_count($Ip, ':')).':', $Ip);     if (strpos($Ip, ':') === 0) $Ip = '0'.$Ip;     return $Ip; }  /**  * Convert IPv6 address to an integer  *  * Optionally split in to two parts.  *  * @see https://stackoverflow.com/questions/420680/  */ function IPv6ToLong($Ip, $DatabaseParts= 2) {     $Ip = ExpandIPv6Notation($Ip);     $Parts = explode(':', $Ip);     $Ip = array('', '');     for ($i = 0; $i &lt; 4; $i++) $Ip[0] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);     for ($i = 4; $i &lt; 8; $i++) $Ip[1] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);      if ($DatabaseParts == 2)             return array(base_convert($Ip[0], 2, 10), base_convert($Ip[1], 2, 10));     else    return base_convert($Ip[0], 2, 10) + base_convert($Ip[1], 2, 10); }   For these functions I typically implement them by calling this function first:  /**  * Attempt to find the client's IP Address  *  * @param bool Should the IP be converted using ip2long?  * @return string|long The IP Address  */ function GetRealRemoteIp($ForDatabase= false, $DatabaseParts= 2) {     $Ip = '0.0.0.0';     if (isset($_SERVER['HTTP_CLIENT_IP']) &amp;&amp; $_SERVER['HTTP_CLIENT_IP'] != '')         $Ip = $_SERVER['HTTP_CLIENT_IP'];     elseif (isset($_SERVER['HTTP_X_FORWARDED_FOR']) &amp;&amp; $_SERVER['HTTP_X_FORWARDED_FOR'] != '')         $Ip = $_SERVER['HTTP_X_FORWARDED_FOR'];     elseif (isset($_SERVER['REMOTE_ADDR']) &amp;&amp; $_SERVER['REMOTE_ADDR'] != '')         $Ip = $_SERVER['REMOTE_ADDR'];     if (($CommaPos = strpos($Ip, ',')) &gt; 0)         $Ip = substr($Ip, 0, ($CommaPos - 1));      $Ip = IPv4To6($Ip);     return ($ForDatabase ? IPv6ToLong($Ip, $DatabaseParts) : $Ip); }   Someone please tell me if I'm reinventing the wheel here or I've done something wrong.  This implementation converts IPv4 to IPv6. Any IPv6 address it doesn't touch. When I run this command :  apt install libkf5*   I get the following error :  Unable to locate package libkf5*    I'm sure that the packages whose names start with libkf5 exist (tab-completion says so) . So the problem is not about the absence of those packages.(Note the asterisk at the end of that error message , the wildcard's not working at all) I use apt v.1.9. Thanks in advance. Change password on root user and user account I know you can export the asset as a PSD and open it in Photoshop. Is there a more seamless way?  Update:  I'm an idiot.  Someone else created the logo, and the shadow was it's own layer on multiply and not a normal drop shadow effect. Would fewer Jedi be more powerful? What does outputting to /dev/null accomplish in bash scripts? Random images appearing in my home directory... Where are they coming from? Here's an odd problem I've had for a while. Every so often, I find that I have several jpeg images sitting in my home directory that I didn't put there. I'm fairly sure this isn't a security breach but rather, there's something writing these files in there that I've forgotten about.  Here's what I'm looking at today:  oli@bert:~$ ls -l *.jpg -rw------- 1 oli oli  8061 2010-11-19 18:30 An-alien-from-Mars-Attack-002.jpg -rw------- 1 oli oli 24362 2010-11-19 18:30 An-alien-from-Mars-Attack-006.jpg -rw------- 1 oli oli  8147 2010-11-05 18:08 Avatar-Film---2009-002.jpg -rw------- 1 oli oli 35602 2010-11-05 18:08 Avatar-Film---2009-006.jpg -rw------- 1 oli oli  6058 2010-12-03 18:42 Britney-Spears-victim-of--002.jpg -rw------- 1 oli oli 28330 2010-12-03 18:42 Britney-Spears-victim-of--006.jpg -rw------- 1 oli oli  8135 2010-11-12 19:19 charlie-brooker-twitter-p-002.jpg -rw------- 1 oli oli 22115 2010-11-12 19:19 charlie-brooker-twitter-p-006.jpg -rw------- 1 oli oli  8059 2010-10-29 18:47 Leonardo-DiCaprio-002.jpg -rw------- 1 oli oli 29988 2010-10-29 18:47 Leonardo-DiCaprio-006.jpg -rw------- 1 oli oli  8064 2010-10-22 16:41 Spot-the-difference-.-.-.-002.jpg -rw------- 1 oli oli 29188 2010-10-22 16:41 Spot-the-difference-.-.-.-006.jpg   They all seem like they're TV or film related... But I honestly don't have a clue what could be writing them. They don't bother me enough to go on a full-scale witch hunt but if anybody knows of something that writes images in the home directory like this, please let me know. I could install with an error message "unknown media /ALL". It could not run WPS Office. Mouse Pointer Locator Due to a stroke, I have had some vision impairment in that I sometimes have a hard time locating my mouse pointer on the screen.  I have tried making the mouse pointer larger which helps but is still not the solution.  I am new to the Mac world and would appreciate your help.  In Windows, there is an option where pressing the CTRL key will help to identify the mouse pointer using a temporary display of several radiating concentric circles around the mouse pointer.  Is there something like this on OS X? I'm trying to debug a Python script by placing a print("foo=" + foo) statement in my code.  But where does the output appear?     It doesn't appear on the Python Console, and it doesn't appear on the area that appears when you drag the lower edge of the infobar downwards (what is the name for this area? My script errors show up in red here.) Can the Next/Prev Post links be ordered by menu order or by a meta key? I have a series of posts that are ordered by a meta_key value. They could also be arranged by menu order, if necessary.  The next/prev post links (generated by next_post_link, previous_post_link, or posts_nav_link all navigate by chronology. While I understand this default behaviour, I don't understand how to change it. I found that it maps through to adjacent_post_link in link-template.php, but then it starts to seem fairly hard-coded. Is it recommended to re-write this from scratch to replace it, or is there a better solution. If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing. Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. Favicon of Earth Science and Spanish are too similar (exactly the same in HNQ list) The favicon is used in various places around the Stack Exchange (SE) network, to identify the site. Currently, Earth Science has  for main and  for meta. The problem I see is that these are awfully similar (i.e. exactly the same, as demonstrated here) to those for Spanish.SE, which are  for main and  for meta. The larger versions offer slight differences, but it's still hard to tell quickly which is which. Power users on this site, did you guess right, even after some time to think about it? (Answers under spoiler tag).   for main and  for meta.  for main and  for meta.    The first one is Earth Science and the second is Spanish.  Should SE visually distinguish between the two very different sites? How to achieve line break in simple $formula$-mode We know that you can't transfigure food, so you have to grow it or buy it. Ditto money.  I'm sure that aside from some medicinal plants Herbology Professor had there were no farmlands at Hogwarts. So they must have bought food (as well as potions ingredients for Snape etc...). And they needed to pay the teacher salaries.  Where did all that money come from according to canon?  Ministry funding?  Tuition?  Patent money on Dumbledore's inventions?  Philanthropic donations by Malfoys and co?  Other? Reinhardt's main strength is his shielding abilities and the protection he provides for his team, but Are there any abilities that cannot be blocked by Reinhardt's barrier field? What 70s (60s?) TV show / TV movie had a generation ship whose inhabitants did not know they were on a spaceship? I have the vaguest memory of a TV show (maybe a BBC import?) in which the protagonists were moving through the sections of a massive generation ship whose inhabitants had forgotten that they were on a spaceship... I know there was an RPG with the same plot (which, after all, is a pretty great blank slate) but I definitely feel I saw this on a static-filled 13" B&amp;W TV back in the day... Context  I'm having trouble understanding the limited situations in which qualifiers can be distributed.  I am given that the rules are:  $$\forall x\left[P(x)\land Q(x)\right]\equiv\forall xP(x) \land\forall xQ(x) \tag{1}$$ $$\exists x\left[P(x)\lor Q(x)\right]\equiv\exists xP(x) \lor\exists xQ(x) \tag{2}$$  $$\forall x\left[P(x)\lor Q(x)\right]\not\equiv\forall xP(x) \lor\forall xQ(x) \tag{3}$$ $$\exists x\left[P(x)\land Q(x)\right]\not\equiv\exists xP(x) \land\exists xQ(x) \tag{4}$$    Questions   What are examples of $P(x)$ and $Q(x)$ that illustrate when these statements are true? How can I show these statements always hold using proofs? Why can't "$\forall x$" be distributed in example 3 or "$\exists x$" in example 4?