If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} - x_{n_{k-1}}\| < 1/2^k$
Show that a Cauchy sequence has a fast-Cauchy subsequence
How to typeset a file path? How do I merge 2 VMware disks in to a single one? I have installed an Ubuntu server on a VMWare workstation with a space of 20GB storage as seen below:    After the installation I've attached another 40GB worth storage to the existing Ubuntu server installation (seen below):    Now I want to add this 40GB to my / (root) directory it mean's  I want that my / directory became to 60G , like it will be resize from 20G to 60G (20G + 40G), i don't know what should i do, can any body help me ?  , my df -h result shows;  root@Userver:~# df -h Filesystem      Size  Used Avail Use% Mounted on /dev/sda1        18G  900M   16G   6% / udev            990M  4.0K  990M   1% /dev tmpfs           400M  308K  399M   1% /run none            5.0M     0  5.0M   0% /run/lock none            999M     0  999M   0% /run/shm none            100M     0  100M   0% /run/user root@Userver:~#   it must be 60G !  i have no graphic mode, then i can't use Gparted tools, i'm just have ssh access ! Share your experiences on exploring iOS 7 with us..  Whether you found hidden or little-known feature of iOS 7 or can explain how a big feature makes a difference, we're looking for objective reports of how the new OS affects your workflow.  Also provide details on how to achieve that feature, and if possible, include a relevant image too (hold Home and Lock for a screenshot on your iOS device).    Please post one tip per answer. Please also check to see if your answer has already been posted - duplicate answers will be deleted in favour of the original answer on any given topic. To search answers for this question use inquestion:101382 (or inquestion:this, directly from the question page) in addition to your search terms in the search box in the upper right hand corner of this page. How to control the mouse in Mac using Python? What would be the easiest way to move the mouse around (and possibly click) using Python on OS X?    This is just for rapid prototyping, it doesn't have to be elegant. Unable to login after password change, having home directory encrypted 16.04 VirtualBox VM (from vhd file) hangs at "non blocking pool is initialized" I'm trying to boot Canonical's Azure 16.04 .vhd file in a VirtualBox VM and it hangs at "non blocking pool is initialized".  It works in Hyper-V, so it may only be a bug in virtual box (5.0.20). How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$? Theorem: Let $f:[0,1] \to [0,1]$. Assume $f$ is continuous. Then there exists   $c \in [0, 1]$ such that $f(c) = c$.         For any function $f$, let $g(x) = f(x) − x$. If the theorem is true, what does that say about $g(c)$?   Write a proof of the theorem. Do it by applying the Intermediate Value Theorem   in the appropriate way to the function $g(x) = f(x) − x$ with $a = 0$ and $b = 1$. Clearly state why the hypotheses of the IVT are satisfied here.        Well if the theorem is true, then there exists $c$ such that $f(c) = c$. Then $f(c) - c = 0$. Then $g(c) = 0$. Intermediate Value Theorem: Let $I$ be an interval and $f$ a function whose domain contains $I$. If $f$ is continuous, then for all $a, b \in I$ with $a &lt; b$ and all real numbers $k$, if $k$ is strictly between $f (a)$ and $f (b)$, then there exists $c$ such that $a &lt; c &lt; b$ and $f (c) = k$.   Proof Attempt: Take $a = 0$ and $b = 1$. Then $a,b \in[0,1]$ and $a &lt; b$. Since $f:[0,1] \to [0,1]$, it follows that for all real numbers $k$ such that $f(a) &lt; k &lt; f(b)$, then there exists $c$ such that $a &lt; c &lt; b$ so $0 &lt; c &lt; 1$ and $f(c) = k$. Then $f(c) - k = 0$. Thus, since $g(x)=f(x)-x$, it follows that $g(c) = 0$.   Looking to see if I did this correctly and/or if there is a more elegant way to prove problem 2. I am a little confused why the first theorem is true, but I went with it. $ echo $(( 255 )) 255 $ echo $(( 33 )) 33 $ echo $(( ~33 )) -34 $ echo $(( ~255 )) -256 $    and my kernel is:  $ uname -a Linux HOSTNAME 3.2.0-40-generic-pae #64-Ubuntu SMP Mon Mar 25 21:44:41 UTC 2013 i686 i686 i386 GNU/Linux     QUESTION: ~ is for negating the number AFAIK. But why does ~33 produce -34 and why does ~255 produce -256? How to add spacing between columns? Calling future method from batch class? Does centrifugal force exist? Currently in my last year of high school, and I have always been told that  does not exist by my physics teachers. Today my girlfriend in the year below asked me what centrifugal force was, I told her it didn't exist, and then she told me her textbook said it did, and defined it as "The apparent force experienced towards the outside of a circle is the centrifugal force and is due to the mass of the object resisting the inward centripetal acceleration that the object is experiencing". I was pretty shocked to hear this after a few years of being told that it does not exist.  I did some reading and found out all sorts of things about pseudo forces and reference frames. I was wondering if someone could please explain to me what is going on? Is it wrong to say that centrifugal force does not exist?   This has always nagged me a bit as I often wonder that if every force has a reaction force then a centripetal force must have a reaction centrifugal force, but when I asked my teachers about this they told me that centrifugal force does not exist. Proof of -(-v)=v in a vector space The exercise is: Prove that $-(-v)=v$ for every $v \in V$  Proof Suppose  $v \in V$ and $V$ is a vector space. Then $-(-v) \in V$ as result of the scalar multiplication property and $-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $ The desired result $-(-v)=v$ holds.  The solution manual gives the proof:  Proof   I just wanted to make sure that the way I did the proof isn't missing anything?Thanks I really like the , where certain questions from the Stack Exchange Network are presented, along with a hotness rating that is described as &quot;arbitrary&quot; in its tooltip. Such questions also appear randomly on the sidebar on sites across the network, under the heading &quot;Hot Network Questions&quot;. How do these arbitrary hotness points work, and how are they used to select questions to appear in those lists? What does "\" mean in math Adobe Illustrator: strange behavior while aligning objects I am creating icons with Adobe Illustrator during which process I use the Align tool quite frequently.  All of a sudden, the align tool started behaving quite strangely.  When I select two objects and specify that I want to align to a key object, when I press align it seems to only align halfway to where it should be.  When I press align again it aligns halfway again - after I press it several more times, it finally gets to where it needs to be.  Also, when I try to align to artboard, it pushes the object out to a seemingly random place.  Here is a video showing exactly what I mean:   What setting did I accidentally set to make align behave like this? I'd like to work on a bbcode filter for a php website. (I'm using cakephp, it would be a bbcode helper) I have some requirement.    Bbcodes can be nested. So something like that is valid.  [block]       [block]       [/block]       [block]           [block]           [/block]       [/block]   [/block]     Bbcodes can have 0 or more parameters.     Exemple:     [video: url="url", width="500", height="500"]Title[/video]   Bbcodes might have mutliple behaviours.     Let say, [url]text[/url] would be transformed to [url:url="text"]text[/url] or the video bbcode would be able to choose between youtube, dailymotion....  I think it cover most of my needs. I alreay done something with regex. But my biggest problem was to match parameters. In fact, I got nested bbcode to work and bbcode with 0 parameters. But when I added a regex match for parameters it didn't match nested bbcode correctly.   "\[($tag)(=.*)\"\](.*)\[\/\1\]"  // It wasn't .* but the non-gready matcher  I don't have the complete regex with me right now, But I had something that looked like that(above).  So is there a way to match bbcode efficiently with regex or something else.  The only thing I can think of is to use the visitor pattern and to split my text with each possible tags this way, I can have a bit more of control over my text parsing and I could probably validate my document so if the input text doesn't have valid bbcode. I could Notify the user with a error before saving anything.  I would use sablecc to create my text parser.   Any better idea? or anything that could lead to a efficient flexible bbcode parser?  Thank you and sorry for my bad english... Why is fundamental group insufficient to classify manifolds? Should I normalize word2vec's word vectors before using them? After training word vectors with word2vec, is it better to normalize them before using them for some downstream applications? I.e what are the pros/cons of normalizing them? How to jump to the region header from the endregion tag in c# visual studio 2012? I worked on a Word document and also saved the file on my flash drive.  When it asked me to save, I pressed the Replace document option by mistake.    I can see the missing work under the Properties menu. Will I be able to recover the lost work?