Show that $|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$ where $a,b,c\in\mathbb{R}$  I'd like to get an hint on how to get started. What I thought to do so far is dividing to cases to get rid of the absolute value. $(++, +-, -+, --)$ but it looks messy. I'm wondering if there is any nicer way to solve it.  Would love to hear some ideas.  Thanks in advance!
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2}  \right |\leq \left |b-c  \right |$. Give me a clue, I don't have any idea to work with that inequality.
I have the following commands.  cd import zcat urls1.sql.gz | mysql -u root -p urls cd /var/www/project1/ nano 1.php   As of now I'm executing it one by one.  Is there a way to combine those commands in one line? Is voting to balance in the spirit of the site? The other day I saw an answer which was ok. Not great but not bad. So no reason to upvote or downvote. However the answer already had a single downvote. I felt this was unfair so upvoted the answer to keep it at what I felt was a fair score. It wasn't until afterward that it occurred to me that I should have perhaps solely voted on the content of the answer.  Is this a reasonable reason for an upvote or should I have left the answer alone? How to make `man` work for shell builtin commands and keywords? Integrating $\int_0^1 \frac {\log(1-x)\log^2(1+x)}x \mathrm{d}x$ Question: Integrate$$\int\limits_0^1dx\,\frac {\log(1-x)\log^2(1+x)}x=-\frac {\pi^4}{240}$$   I'm curious as to if there is a way to integrate this. I've tried using integration by parts to get$$I=-\frac {\pi^2}6\log^22+2\int\limits_0^1dx\,\frac {\operatorname{Li}_2(x)\log(1+x)}{1+x}$$However, I'm not sure how to continue even further. The polylog in the second integrand seems a bit intimidating and I don't see how the first term even helps. Why did Mann's base explode? After the scuffle between Mann (Damon) and Cooper (McConaughey) in Interstellar, our protagonist gets a hold of a radio and alerts his shipmates to Mann's duplicity. Then his ship explodes. What happened there? Can helium disappear from Earth? In the android app, looking at bounty details using the i button, the bounty reasons are not shown:        The bounty message can provide important information as to the setter's requirements (who may not be the OP and therefore does not wish to edit the question to do so). How prove :         $$ {31}^{11}&lt;{17}^{14} $$       help please! How to ask great questions As programmers we are daily faced with having to get requirements, test things out, etc.  It's practically half our job to know what questions to ask, and how to ask them to get the most concise, actionable answer possible, especially from reluctant users/managers/customers/etc.  How do you ask questions that get answers? Generating random results by weight in PHP? I know how to generate a random number in PHP but lets say I want a random number between 1-10 but I want more 3,4,5's then 8,9,10's. How is this possible? I would post what I have tried but honestly, I don't even know where to start. Show images related to features in QGIS? Say for example, if there's a point data with old trees, then when you use the identify tool, you can see the photograph of the tree  Or even one level further, the labels displaying these images right away without any clicking needed  I believe the term is file streaming? If particles are points, then aren't atoms empty space? Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing? Where do the American and Japanese Transformers franchises meet? The original Transformers TV series was created by the American company Hasbro, and was adapted into Japanese as well. At some point, the Japanese franchise split off into its own continuity, launching its own set of sequels and spin offs.  The questions is: What part of the American franchise is considered canon by the Japanese one? Minor details such as name changes are not to be taken into consideration. Any Info on Generalized Binomial and Exponential Series? I was reading Concrete Mathematics by Knuth et al. and on p. 200, it references the generalized binomial series $\mathcal B_t(z)$ and the generalized exponential series $\mathcal E_t(z)$, defined as $$\mathcal B_t(z)=\sum_{k\ge0}(tk)^{\underline{k-1}}\frac{z^k}{k!}\quad\text{and}\quad\mathcal E_t(z)=\sum_{k\ge0}(tk+1)^{k-1}\frac{z^k}{k!},$$ where $x^{\underline k}$ denotes the falling factorial. I was interested in the generalized binomial series because it satisfied the identity $$\frac{\mathcal B_t(z)^r}{1-t+t\mathcal B_t(z)^{-1}}=\sum_{k\ge0}\binom{tk+r}{k}z^k.$$ I've been trying to find a way to compute the sum $\sum_{n\le k\le m}\binom{tk+r}{k}$, so this seemed related. It seems that $\mathcal E_1(z)$ is also somewhat related to the Lambert $W$ function, though I might be wrong. Searching them up online didn't seem to get anywhere, and I just want to find more sources where I can learn about these functions. If anyone can find some resources, I'd appreciate it! How do I shrink the partition of an img file made from dd? Movie/TV series with a scene of spiky, black goo growing out of a lab container? There was a scene from a sci-fi horror video, not animated, don't know if it's a movie or TV series, where there was this black goo in a glass container and some scientists were looking at it under a microscope.   The pitch black goo grew and grew so fast, eventually breaking out of the inner box and the scientists that were inside the "inner lab" were begging to get out but the scientists outside that inner lab refused to open the door. The black goo was spiky (had spikes).  It's not any of the "Alien" movies, "The Blob", "Life", or "The Thing". It's not vintage. It was made after 1990, I'm pretty sure. For some time, Microsoft has supported an option to remove the limit of 260 symbols for NTFS file name length.  The web is full of articles, describing what should be done.   Hit the Windows key, type gpedit.msc and press Enter. Navigate to Local Computer Policy > Computer Configuration > Administrative Templates > System > Filesystem > NTFS. Double click the Enable NTFS long paths option and enable it.   However, I don't have such option. Please see the screenshot below:    Am I missing some Windows update, or is it something else? What is the difference between a function decorated with  and one decorated with ? Deleting a channel on slack.com How do I enable Scroll Lock?