Prove: $|x-y|\leq |x|+|y|$ Can you help me to prove that $$|x-y|\leq |x|+|y|$$ I get a proff of this equality, but it's very short and I don't know if it's correct.
Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
Reset Windows 10 from USB If $\sum_{1}^{\infty}(a_n)^3$ diverges, does $\sum_{1}^{\infty}(a_n)$? Per the title, if $\sum_{1}^{\infty}(a_n)^3$ diverges, does this imply that $\sum_{1}^{\infty}(a_n)$ diverges?   I'd appreciate hints (!) for dealing with this excercise.  EDIT Per the contrapositive, it is not given that $a_n$ converges absolutely, or that it is nonnegative for all $n$.  Thank you! Is there a tool to encrypt a file or directory? What's the most popular way to encrypt individual files or folders? What does the [Flags] Enum Attribute mean in C#? From time to time I see an enum like the following:  [Flags] public enum Options  {     None    = 0,     Option1 = 1,     Option2 = 2,     Option3 = 4,     Option4 = 8 }   I don't understand what exactly the [Flags] attribute does.  Anyone have a good explanation or example they could post? I'm using latex on a small netbook, and with average-sized files (~150 pages at the moment) compilation is already pretty slow. So I am looking for every way to speed up the compilation.  In this search, I found  explaining how to "precompile a preamble", or, latexically-speaking, how to make a custom format file. That works fine enough.  But my preamble isn't as static as it should be, I keep adding or modifying definitions in there. To mix the two solutions, at the moment, I use a Makefile together with latexmk, the makefile handling the regeneration of the .fmt file if needed, and latexmk taking good care of the rest.  But I have two problems with this :   integration with emacs to compile the whole document when working on a single file (make needs to be called in the proper directory) latexmk doesn't recompile the file every time the preamble is modified, because it doesn't count as a dependency.   So I'd like to find a way to get rid of this Makefile, and get all the stuff done by latexmk.  I edited my .latexmkrc to include this :     pdflatex = 'pdflatex -fmt main %O %S';  add_input_ext('pdflatex','fmt');  add_cus_dep('tex', 'fmt', 1, 'compilepreamble'); sub compilepreamble {     print "Preamble compiling...\n";     $command = '&amp;.pdflatex ./fmt/preamble.tex\dump';     system("pdflatex -ini -jobname='$_[0]' '$command'"); };      But with this, if the file main.fmt is not present or if fmt/preamble.tex is updated, the main.fmt file won't be regenerated. Actually, in the first case, I get an error from pdflatex saying the it can't find main.fmt format file. So I understand latexmk doesn't even try to build main.fmt, it's probably not a problem in my custom dependency. However, I think I've added everything necessary for this to work.  So... Did I forget something? Is there a way round? Or is it a hopeless quest? “Screen overlay detected” dialog? Turned off everything, doesn't work I started receiving the “Screen overlay detected” dialog when trying to change apps settings.  Tried to turn off all "draw over other apps" as it says here - , first without system and then with, but it still doesn't work.  Then, I restored apps to original settings and it just made a bigger mess, dialog continue appearing. Any way to go back to previous settings???  Thanks! Parallel execution of bash script I have a table URL_Experiment in my database (mySQL database). I have 2 million URL links in this table.  For each URL, I am checking if some particular text is present in the URL and updating the information back in my URL_Experiment table.    Now, I need to make the script run fast. The easy solution that comes to my mind is:  split the URL_Experiment into 10 tables (each with 200,000 rows) and run the script 10 times simultaneously. This way, I can ensure that the updates happen properly. But for this, I need to save 10 scripts, each accessing the correct table (I need to rename the tables from 1 to 10).   Though the above seems a pretty neat solution, I am looking for a more sophisticated solution. Something which will execute the same script simultaneously (in parallel) as 10 processes. My concern is the execution time. I do not want the resources to be wasted when they can be utilized. What is "superuser" and how do I get that privilege? Watched screen recording but it's not saved to the desktop I recently recorded a session with my dad. I stopped the recording after the session ended (with the setting set to save to Desktop), clicked the icon that appeared on the lower right corner to open the file, and played back a few minutes to make sure it worked. I then clicked Share to send it via AirDrop to my computer, which did not work; the status remained &quot;cancelled&quot; despite my efforts. I assumed I'd be able to access it through my phone by sending it through a third app, so I pressed cancel on the AirDrop pop-up window. The entire screen recording closed and (apparently) vanished forever after the action. I'm extremely frustrated because I can't seem to locate this video. Operator Exponential $e^A e^B = e^{A+B}$ This question comes from an exam in my functional analysis class.  Suppose $X$ is a Banach space, and $T \in B(X,X)$ is a bounded linear operator on $X$. For any non-negative integer $n$, let $$S_n=\sum_{k=0}^n \frac{1}{k!} T^k$$ where $T^k$ is the composition of $T$ with itself $k$ times and $T^0=I$.  We can show that for any integer $k&gt;0$, $\Vert T^k \Vert \le \Vert T \Vert ^k$. Then we can show that $S_n \in B(X,X)$ and there is some $S \in B(X,X)$ such that $S_n \to S$. We write $S = e^T$ for this operator.  Finally, We were asked to show that $e^T$ has an inverse, and that it is $e^{-T}$. My thought is to prove the following claim first: if $A,B \in B(X,X)$ and $AB = BA$, then $e^A  e^B = e^{A+B}$. If the claim is true, it follows that $e^T e^{-T}=I$.  The claim can be proven provided that the product series can be computed with the Cauchy rule.  $$e^Ae^B=\sum_{i=0}^{\infty}\frac{A^i}{i!}\sum_{j=0}^{\infty}\frac{B^j}{j!}=\sum_{k=0}^{\infty}\sum_{l=0}^{k}\frac{A^lB^{k-l}}{l!(k-l)!}$$ $$=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{l=0}^{k}\frac{k!}{l!(k-l)!}A^lB^{k-l}= \sum_{k=0}^{\infty}\frac{1}{k!}(A+B)^k= e^{A+B}$$  But why can the product series be summed in the Cauchy way? I know for real-number series, by Cauchy's theorem, if $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are absolutely convergent to $A$ and $B$, respectively, then we can add $a_i b_j$ in any way, and the resulting series will converge to $AB$. Does this proposition still hold for commutable operators? (It would be greatly appreciated if ideas of proof or reference is suggested.) Scale-dependent displaying of a layer using OpenLayers plugin? When I am creating a layer in QGIS the layer position/lines does not stay the same when I zoom in closer than, it seems like 1: 1000. I am adding 2 images to explain the problem. The moment I zoom in closer than 1: 1 000 then the original layer does not display correctly anymore.  Please, can you assist in this matter? @ variables in Ruby on Rails How to edit files in a terminal with nano? mysql select from n last rows I have a table with index (autoincrement) and integer value. The table is millions of rows long.  How can I search if a certain number appear in the last n rows of the table most efficiently? How do I install openjdk 7 on Ubuntu 16.04 or higher? I would like to upgrade my main system to 16.04, but I work on projects that require OpenJDK 7.   Apparently it is not available from a trivial apt-get install openjdk-7-jdk. Only versions 8 and 9 are listed in the repository.  Can anyone point me to instructions on how to install it? For an architectural visualization, I would like to use MakeHuman models, but:   Only get them to render white with a black outline (on reflection surface too, if possible). Interact with light like normal human bodies in the scene (cast shadows, etc).   I can do the outline in freestyle, but any in-Cycles solution would be appreciated (OSL?) Can Kakashi still use Susanoo after losing his Sharingan? When Kakashi had double Sharingan, he was able to use Susanoo. However, he lost the Sharingan.  Can he still use Susanoo without Sharingan now? Prettify OS X console by coloring output? OS X terminal is not using colors for highlighting output from different commands, for example ls.   When compared with other systems like Ubuntu, this is a big loss of accessibility/productivity.  What solutions do I have to improve this experience? For the moment I found only this guide for  but I am looking for a more complete solution. Does Class need to implement IEnumerable to use Foreach This is in C#, I have a class that I am using from some else's DLL. It does not implement IEnumerable but has 2 methods that pass back a IEnumerator. Is there a way I can use a foreach loop on these. The class I am using is sealed. How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero? I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation.