I'm supposed to show how if $||\cdot||$ is a norm on vector space $X$ then $||x|| - ||y|| \leq ||x-y||$ for any two vectors $x,y \in X$ The properties I have for a norm are  $||x|| &gt; 0$ if $x \neq \vec{0}$ $||\alpha x|| = |\alpha|||x||$ for any $\alpha \in \mathbb{R}$ $||x+y|| \leq ||x|| + ||y||$ for any $x,y \in X$  Here's what I did. $$ \begin{align*}     ||x-y|| &amp;=    ||x   +     (-y)|| \\             &amp;\leq ||x|| +     ||-y|| \\             &amp;=    ||x|| + |-1|||y|| \\             &amp;=    ||x|| + ||y|| \end{align*} $$ Which is certainly true but not what I'm after. Working the other direction I got the following. $$ \begin{align*} ||x|| - ||y|| &amp;= ||x||-||y|| \\               &amp;= ||x|| + (-||y||)  \end{align*} $$ But I believe that's a dead end... Finally I tried multiplying by -1. $$ \begin{align*}   ||x-y|| &amp;\leq ||x|| + ||y|| \\  -||x-y|| &amp;\geq -||x||-||y|| \\   ||x-y|| &amp;\geq -||x||-||y|| \end{align*} $$ But then then I can't change the sign on the $-||x||$ because it may be a large enough increase to make the inequality not true... I feel like this is supposed to be an easy question so I'm embarassed I can't figure it out yet. Help would be appreciaed.
I have already proved Triangle Inequality $\lVert v+w\rVert \le \lVert v \rVert+\lVert w \rVert$ in vector space $\mathbb{R}^n$. However, I'm having difficulties proving the Reverse Triangle Inequality in the vector space.  I have started with   $$\lVert v \rVert^2=v \cdot v-w\cdot w+w\cdot w \le \lVert v-w \rVert+\lVert w \rVert^2$$ $$\lVert w \rVert^2=w \cdot w-v\cdot v+v\cdot v \le \lVert w-v \rVert+\lVert v \rVert^2$$  based on the Triangle Inequality but I don't know if it is correct. If this is right, I think I can go on with proving it the same way I would the normal Reverse Triangle Inequality.
Increasing the size of tyres on a hybrid We are planning on doing some mountain biking in a few weeks time and we'd like to get some wider tyres for my wife's hybrid. I know there are similar questions and answers out there on this topic, but I need some specific help to make sure I buy the right tyres.  The bike is a Ladies Scott Sportster 50 2015, the best website I could find with a fairly accurate spec is     It's currently got Kenda K-180 700 x 38c / 30TPI tyres on it, which are fairly good all rounders and capable of running on clean forest paths but I don't think they would handle anything heavier.  It's got disk brakes so there isn't a clearance issue there, but I have noticed the frame is generally narrower than true MTBs, here are some photos i took of it.    Has anyone put larger, knobblier tyres on this kind of hybrid? whats the largest I can get away with? - I'm also slightly confused by all the tyre sizing, I'm sure someone told me this is a 27.5" tyre, but 700 seems to mean 29" on a lot of websites.  Any advice is very much appreciated, Thanks. Same as title, want to kill alt-tab, etc. I am trying to prove that:  Let $f$ be analytic in the closed unit disc with $f(z)\leq 1$ for $|z|\leq1$.  For a given $0&lt;r&lt;1$ and $|z_1|,|z_2|\leq r$, $z_1\neq z_2$ then,   $$\left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|\leq \frac{1}{(1-r)^2}.$$  My attempt is: part a  of the question asks to prove $$\left|\sum_k^\infty a_nz^n\right|\leq \frac{ |z^k|}{1-|z|}.$$   So, \begin{eqnarray*}  \left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|&amp;=&amp; \frac{|\sum_1^\infty a_nz_2^n-\sum_1^\infty a_nz_1^n|}{|z_2-z_1|}\\  &amp;\leq&amp; \frac{\frac{ |z_1 |}{1-|z_1|}+\frac{ |z _2|}{1-|z_2|}}{|z_2-z_1|}\\  &amp;\leq&amp;\frac{1}{(1-r)^2} \frac{   |z_2| ({1-|z_1|})+  |z _1| (1-|z_2|) }{|z_2-z_1|}\\ \end{eqnarray*} But could not prove that the last term is controlled by 1. What are the mechanics of time dilation and length contraction? Going beyond the mathematical equations involving light and the "speed limit of the universe", what is observed is merely a phenomenon and not a true explanation of why time dilates or length contracts.  It has been proven to work out, but do we know why?  Is it something that happens at a subatomic level? It is quite easy to show that for every prime $p$ and $0&lt;i&lt;p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the numerator is divisible by $p$ whereas the denominator is not (since it is a product of numbers smaller than $p$ and $p$ is prime).  My problem is with generalizing this argument for $q=p^n$. I'm looking for the most elegant and simple way to prove that $p$ still divides $\large q\choose i$. On really old Ubuntu 6.06 - How to upgrade right (opposite of left) connected to right (legal term)? Product of an exponential function of IDD variables if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! How can I send single file to multiple remote sites at the same time? Can scp be used to send a single file to multiple remote servers at the same time? If so, how? If not, what's the alternative? Randomize a List What is the best way to randomize the order of a generic list in C#? I've got a finite set of 75 numbers in a list I would like to assign a random order to, in order to draw them for a lottery type application. Why is this command not working: "ps aux | grep xscreensaver" I didn't see any hits about this on google, so I'm asking you:  I just tried to run this command in an ssh session, local is Debian Wheezy KDE, remote is Fedora 21 Gnome:  ps aux | grep xscreensaver   and this is the output:  bash:  grep: command not found...   I never seen this before, what is the cause?  I just few hours before updated the system, but didn't see any packages marked for removal. There are only two users on the system, and only me installs or removes software on it.  EDIT #1  Here is the commands and output copied,notice that the space is not there the second time:  [root@Hostname ~]# ps aux |  grep xscreensaver bash:  : command not found... [root@Hostname ~]# ps aux | grep xscreensaver bash:  grep: command not found... different font while using url I'm having troubles while using the url command in my bibliography.  I mean:  1) \bibitem{[8]} {\url{http://projects.webappsec.org/w/page/13246989/Web20Application20Security20Statistics\#Tableofcontent}}  2) \bibitem{[5]} Justin~Clarke ``SQL Injection Attacks and Defense'', Syngress Publishing, Inc.2009   Unfortunately font 1) is different from 2).  What should I do in oder to get the same font for both items?   I'd like the font used in 1)  to be identical to the one displayed in 2) Google Chrome PPA upgrade invalid signature Storing Images in DB - Yea or Nay? So I'm using an app that stores images heavily in the DB. What's your outlook on this? I'm more of a type to store the location in the filesystem, than store it directly in the DB.  What do you think are the pros/cons? I have a cryptography workshop question I'm having trouble with as follows;  Person A creates a cipher $E_k(m)$ which produces a ciphertext from message $m$ using key $k$. The function inside $E$ is kept secret but the length of $E_k(m)$ is known.  Person B recommends "increasing" security of the cipher by instead doing :   $(E_k(m) \oplus m)\ ||\ (E_k(m) \oplus 1111...11)$      Does this in fact increase security of the cipher or increase new problems.  My thinking is, depending on the function within $E$, xoring the output of the cipher with the plaintext message $m$ could expose the key $k$, meaning the extra complexity is for nothing. Am I on the right track, or am I missing something?  I have tried searching for examples of similar schemes and found nothing (which probably means it's not a good scheme) but I need to justify my answer.   Any steers in the right direction would be greatly appreciated, I'm more than happy to do the research myself just unsure what specifically to look for.   Unfortunately the above context is all I have been provided for this question. Problem in solving functional equation $f(x^2 + yf(x)) = xf(x+y)$ To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation  $$f(x^2 + yf(x)) = xf(x+y)$$  It can be easily seen that the identity function $i.e.$ $f(x)=x$ and $f(x)=0$ (verified just now) satisfies the above relation!! And putting $y=0$ I have got $f(x^2)=xf(x)$.  Help needed to find other functions satisfying the relation. Should I always use a comma after "e.g." or "i.e."? My wife has a Nikon D3100 that I've been using to film, but the battery typically won't last through long shoots.  I've gathered from searching around that I can't charge the camera with USB, but I wondered if there's ANY way to do that. 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