I am trying to prove that $d(x,y) = |\frac{1}{x}-\frac{1}{y}|$ is a Metric. I already managed to prove:  $d(x,y)\geq 0$ $d(x,y)\geq 0$ iff $x=y$ $d(x,y) = d(y,x)$  But I am stuc at the 4th property:  $d(x,y) \leq d(x,z) + d(z,y)$ (The triangle inequality)  Thus far I came to: $|\frac{1}{x}-\frac{1}{z}| + |\frac{1}{z}-\frac{1}{y}| = |\frac{1}{z}-\frac{1}{x}| + |\frac{1}{z}-\frac{1}{y}|$ But now I am stuk. I understand that this is probably a basic question but I am new to the world of Real Analysis.
I'm trying to prove that  $$ \left| \frac{1}{x}-\frac{1}{y} \right| $$  is a metric on $X = \mathbb R^+$.  The properties of non-negativity and symmetry are obvious (do I actually even need to prove them?), but the tricky one is, as usual the inequality property, i.e.  $$ d(x,z) \le d(x,y) + d(y,z) \text{ for any } z \in X. $$  So, ok, I have that  $$ d(x,z) \le d(x,y) + d(y,z) \\ \iff \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| \\ \iff \left| \frac{1}{x} \right|- \left| \frac{1}{z} \right| \le \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| $$  ...and that's about as far as I've gotten–which is to say, not very far. I also tried multiplying both sides by $ \left| \frac{1}{x}-\frac{1}{z} \right| $, but that wasn't very fruitful. If someone could point me in the right direction, it'd be greatly appreciated. Thanks.
I have recently begun self-studying Number Theory, and am working on proving:     Show that every integer $n&gt;0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and $c_m \neq 0$.   I believe I have already shown the existence portion of this proof correctly, and now I am looking for some hints on the uniqueness part. So far what I have tried is:     Suppose for the sake of contradiction that there is an alternate representation of $n$ besides $n = \sum_{i=0}^mc_i3^i$, say $n = \sum_{i=0}^pb_i3^i$ where we still have $b_p \neq 0$ and $b_i \in \{-1,0,1\}$.    At this point I felt that I first need to establish that $m=p$, and second show $c_i=b_i$ for each $i \in \{1,2,\dots,  m\}$. To show the former, I left off in my proof by contradiction with      Without loss of generality suppose $p&gt;m$. We know there is an integer $q$ such that $m+q=p$. We have two ways of writing $n$, which means $$ \sum_{i=0}^pb_i3^i-\sum_{i=0}^mc_i3^i=0 \\ \implies \left(\sum_{i=0}^mb_i3^i+\sum_{i=m+1}^{m+q}b_i3^i\right)-\sum_{i=0}^mc_i3^i=0 \\ \implies \sum_{i=0}^m(b_i-c_i)3^i+\sum_{i=m+1}^{m+q}b_i3^i=0 \\ \implies \sum_{i=m+1}^{m+q}b_i3^i=\sum_{i=0}^m(c_i-b_i)3^i$$   At this point I am stuck. If anyone has any hints to help me find the contradiction or feel that there is a better way to establish uniqueness, please let me know! Are these intended activities legal to do in the USA under the VWP? I'm a British citizen traveling to the USA next month with an ESTA to see my fiancee. I'm traveling there to plan our wedding, talk about a location to get married, and probably finalise a date and a booking with my fiancee (US citizen). This is the primary reason for my trip to the USA. Am I allowed to do this as a tourist? I would like to know for when the CBP officer asks me if this thing is allowed.  Am I allowed to enter the USA, on a separate trip (probably later this year), for the sole purpose of marrying my fiancee, provided I can persuade the CBP officer I intend to leave at the end of my trip? My fiancee and I are moving to the UK next year. Irrationals: A Group? I understand that the set of irrational numbers with multiplication does not form a group (clearly, $\sqrt{2}\sqrt{2}=2$, so the set is not closed). But is there a proof or a counter-example that the irrationals with addition form (or do not form) a group?  Thank you!  Edit: In particular, I am wondering if the set is closed with respect to addition. Real analysis derivate problem. How can I prove this? Let $f$ be differentiable in $[a,b]$ and $f(a)=0$. If $\exists M \in R$ such that $\vert f'(x) \vert \leq M \vert f(x)\vert \ $,  $\forall x \in [a,b]$, then $f(x)=0, \ \forall x \in [a,b]$.  $\\$ I have an idea, but I can´t see it clear. I have arrived to the inequality $\vert f(x)\vert \leq  K\vert f(y)\vert $ for $y \in (a,x)$, supposing $k&gt;0$ and repeating the process $\vert f(x)\vert \leq  K^n\vert f(z)\vert$ for $a&lt;z&lt;y&lt;x$. If $K&lt;1$, then if $n \rightarrow \infty \ $, $f(x)=0$. That´s a sort of intuitive idea. I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. $[K(\zeta_n):K]=\phi(n)$? Is it true that $[K(\zeta_n):K]=\phi(n)$ where $\zeta_n$ be primitive root of unity and K be field of char zero?  I think it should be equal to the degree of cyclotomic polynomial which has degree $\phi(n)$.  Can someone please elaborate that what is the procedure for finite fields build folder usage issues In an ideal case, we assume strings and ropes to be massless and having same tension throughout.  However, I was doing some questions which involved a rope with mass.  The question was that if a uniform thick rope of length 5 m is resting on a horizontal smooth (assume frictionless) surface and is pulled by a force of 5N from one end, then what is the tension in the rope at the point 1 m away from the end where force is applied? (sorry for the poor image)    I came to the conclusion that since rope is not fixed, tension anywhere should be zero. However, when I checked the answer, it was 4 N. I failed to understand how.  Then the next part of the question was that if the rope now hangs between two fixed points which are at same level, then what would be the tension in the rope at its lowest point?    The answer has to be expressed in terms of mass of string m, acceleration due to gravity and $\theta$ where $\theta$ is the angle that tangents to the rope make with the horizontal.  I took that force acting on lowest point is mg, since it is center of mass and hence T = mg. However, according to the answer it depends upon $\theta$ as well. However, I failed to see what I was doing wrong?  I would appreciate any help in solving these questions, since I have no clue of where exactly I am wrong. I did search the Internet and a few books, but everywhere they assumed string/rope to be massless. I want my Java code to be typeset in a monospaced font, so I am using the verbatim environment. LaTeX commands are highlighted in Eclipse, but as soon as I use the verbatim environment everything inside and the color of the environment itself becomes gray.  \begin{verbatim}   public class main {  public static void main (string[] args){      for(int i = 0; i &lt; arg.length; i ++){          System.out.println(args[i]);      }    }  };  \end{verbatim}   I'm using the verbatim and alltt packages.  Update 1.  MWE.  \documentclass[xcolor=dvipsnames]{beamer}  \mode&lt;presentation&gt; { \usetheme{CambridgeUS} \setbeamercovered{transparent} }  % Поддержка русского языка \RequirePackage[utf8]{inputenc} \RequirePackage[T2A]{fontenc} \RequirePackage[russian]{babel}  \RequirePackage{amsfonts}  \usepackage{algorithmic} \usepackage{mathptmx} \usepackage[scaled=.90]{helvet} \usepackage{courier} \usepackage{pst-node} \usepackage{pst-tree} \usepackage{auto-pst-pdf} \usepackage{color} \usepackage{colortbl} \usepackage{pstricks} \usepackage{verbatim} \usepackage{alltt} \usepackage{listings} \usepackage{animate} \usepackage{multimedia}  \title{Title}  \author{Name}  \institute{}  \date{\today}  \begin{document}  \begin{frame} \titlepage \end{frame}  \begin{frame} \frametitle{Outline} \tableofcontents \end{frame}  \section{Introduction}  \begin{frame}{Классификация по поколениям}  \begin{itemize}  \item Третьего поколения (высокоуровневые языки) - последовательность высокоуровневых конструкций (циклы, условные выражения, классы, интерфейсы, методы, процедуры, \ldots) \pause  \begin{exampleblock}{Пример}  \begin{centering}  Fortran, C, C++, C\#, Java, Lisp.  \begin{verbatim}  public class main { public statiс void main (string[] args){ for(int i = 0; i &lt; arg.length; i ++){ System.out.println(args[i]); } } };  \end{verbatim}  \end{centering}  \end{exampleblock}\pause  \item Четвертого поколения (для конкретного применения) - последовательность инструкций конкретной предметной области \pause  \begin{exampleblock}{Пример} SQL, Postscript, XML, Mathlab. \end{exampleblock} \pause  \end{itemize}  \end{frame}  \end{document} Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks Defining my own None-like Python constant The Definitive C++ Book Guide and List This question attempts to collect the few pearls among the dozens of bad C++ books that are published every year. Unlike many other programming languages, which are often picked up on the go from tutorials found on the Internet, few are able to quickly pick up C++ without studying a well-written C++ book. It is way too big and complex for doing this. In fact, it is so big and complex, that there are very many very bad C++ books out there. And we are not talking about bad style, but things like sporting glaringly obvious factual errors and promoting abysmally bad programming styles. Please edit the accepted answer to provide quality books and an approximate skill level — preferably after discussing your addition in . (The regulars might mercilessly undo your work if they disagree with a recommendation.) Add a short blurb/description about each book that you have personally read/benefited from. Feel free to debate quality, headings, etc. Books that meet the criteria will be added to the list.  Books that have reviews by the Association of C and C++ Users (ACCU) have links to the review. *Note: FAQs and other resources can be found in the  and under . Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity Can anyone explain me please how to see this statement: the only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the Identity.  how can I prove formally that? or how can I see that only the Identity satisfy it? I have a table that is very wide. Now I want to force it into one landscape page. I already set the page orientation to landscape but now I don't know how to force it into one page with scaling. Any hints? What is the benefit of breaking up a continuous predictor variable? I'm wondering what the value is in taking a continuous predictor variable and breaking it up (e.g., into quintiles), before using it in a model.   It seems to me that by binning the variable we lose information.   Is this just so we can model non-linear effects?  If we kept the variable continuous and it wasn't really a straight linear relationship would we need to come up with some kind of curve to best fit the data? Is it safe to use a voltage divider for Arduino and DC motor controller? I attempted to log out with the following actions:    This accomplished absolutely nothing since I remained logged in after clicking "log out" on the confirmation page. I then tried logging out of Stack Exchange from one of the network sites and repeated the steps above. The results were the same. I remained logged in no matter which combination of steps I performed.  In fact, the only way I could log out at all was to clear all of the cookies that were set. System freezes completely with Intel Bay Trail My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540) I have the following validation rule:  AND(  $Profile.Name &lt;&gt; 'System Administrator',  OR(  ISPICKVAL(PRIORVALUE(Level__c), "Archived"),  ISPICKVAL(PRIORVALUE(Level__c), "Closed")  )  )   This validation rule is on the opportunity object, and what I want to achieve is, if the picklist Level__c is either set to Archived or Closed, the opportunity record can only be edited by the System Administrator.  For some reason though, the validation rule also triggers, when I want to set the Level__c to Archived or Closed. But this is not what I want.   Any idea what's wrong with the validation rule? Can the 5E abjurer wizard recover temp hp on his arcane ward any time it is below max or only when it reaches 0? On p115 of the PHB under Arcane Ward there are a few facts that I think are clear:   The ward is created by the first abjuration spell you cast in a day It has a max hp of level &times; 2 + int modifier. The ward does not absorb damage once it reaches 0 hp, but it remains until you take a long rest The ward can be recharged by casting abjuration spells, at a rate of 2 x spell level worth of hp 3.   However, the way they wrote the paragraph about recharging seems unclear to me. Can you recharge the ward whenever it is below max hp or only when it is at 0? My understanding of temporary hp would say it is the latter, but the writing on this ability infers the former.  An example.  If I were a level 10 abjurer with 20 int and cast Mage Armor, it would give the ward 25 hp. If I took 30 damage from an enemy, the ward would absorb 25 damage, leaving it at 0, and I would take the remaining 5 damage. If I cast Stoneskin (level 4 abjuration) the ward would be recharged with 8 hp. If I then cast Planar Binding (level 5 abjuration) which would be correct?   The ward remain at 8 hp The ward would recharge an additional 10 hp to a total of 18 hp