I was wondering, Is there Cauchy sequence that are not bounded ? Of course, in complete spaces is not possible. I have a theorem that says that if $(A,d)$ is not complete, then $(\bar A,d)$ is complete. But are there spaces s.t. indeed for all $\varepsilon&gt;0$, there is $N$ s.t. $d(x_n,x_{m})&lt;\varepsilon$ for all $n\geq N$, and all $r\geq 0$, but the ball also move to $+\infty $ ?
Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I'm not sure if it's totally correct. It's quite short though. I was just hoping someone could look it over and see if it is a valid proof. Thank you!  Lemma: Every Cauchy sequence is bounded.  Proof: Let $(a_{n})$ be Cauchy. We choose $ 0&lt;\epsilon_{0}$. So $ \forall \; n&gt;m\geq N_{0}$ we have that $\vert a_{n}-a_{m} \vert &lt; \epsilon_{0}$. Therefore $(a_{n})$ is bounded for all $ m \geq N_{0} $ by $ \epsilon_{0} $. Since $ \mathbb{N}_{N_{0}}$ is finite, it is bounded. So, for all $ m&lt;N_{0} $, $ (a_{n})$ is bounded. Therefore $(a_{n})$ is bounded.  I realize I haven't said what the bounds are but I think that's sort of irrelevant. So long as we know it's bounded. Any help is much appreciated!
Limit of a stochastic integral Formulae for Catalan's constant. Some years ago, someone had shown me the formula (1). I have searched for its origin and for a proof. I wasn't able to get true origin of this formula but I was able to find out an elementary proof for it.  Since then, I'm interested in different approaches to find more formulae as (1).  What other formulas similar to ($1$) are known?  Two days ago, reading the book of Lewin "Polylogarithms and Associated Functions" I was able to find out formula (2).  $\displaystyle \dfrac{1}{3}C=\int_0^1 \dfrac{1}{x}\arctan\left(\dfrac{x(1-x)}{2-x}\right)dx\tag1$   $\displaystyle \dfrac{2}{5}C=\int_0^1 \dfrac{1}{x}\arctan\left(\dfrac{\sqrt{5}x(1-x)}{1+\sqrt{5}-\sqrt{5}x}\right)dx-\int_0^1 \dfrac{1}{x}\arctan\left(\dfrac{x(1-x)}{3+\sqrt{5}-x}\right)dx\tag2$   $C$ being the Catalan's constant.  I have a proof for both of these formulae.  My approach relies on the following identity:  For all real $x&gt;1$,  $\displaystyle \int_0^1 \dfrac{1}{t} \arctan \left (\dfrac{t(1-t)}{\frac{x+1}{2}-t}\right) dt=\int_1^{\frac{\sqrt{x}+1}{\sqrt{x}-1}}\dfrac{\log(t)}{1+t^2}dt$ In the story I'm trying to recall, everybody on their thirteenth birthday (?) is assigned by some kind of brain analysis method a profession and given some kind of pill (?) that instantly (?) grants them the required knowledge and skills to start working immediately. Some professions are more sought after than others, especially in view of the olympic (?) games regularly held on a profession-by-profession basis.  The protagonist, however, is turned down from this system and dismissed with some general reason and has to work his way to adulthood the boring old way: studying from books. Eventually he rejects from his status as a "special" student and goes on the run, hiding from the teachers and envying the normal people.      Eventually, it turns out that the protagonist was actually smart enough that being spoonfed premade information was a waste. What he actually was assigned to was advancing the bundles of knowledge themselves, as well as being front line in the technological progress.  "Somebody has to write those textbooks! You can infuse knowledge about how to use today's technology, but you can't do that for tomorrow's - people that get assigned a profession become tied with their assigned technology and are only as relevant as that. If you want more, you must learn how things actually work, and there's no shortcut for that." (not an actual quote, but a specific passage as I remember it.)  The troubles the protagonists underwent were actually planned and necessary for him to acknowledge and accept his role in society.   This story was certainly longer than a couple pages, but it wasn't very long either. If it's a short story, it was probably written by Asimov, but I can't be sure. It rings me of Brave New World given the theme of government-issued population planning, but it certainly doesn't fit with anything else in the actual plot. Any idea? I'm doing a masters course at a different university than I did my undergrad (both in maths). I did well in my undergrad but now I'm finding it very difficult. The shock of the change is hard to cope with.  I'm not just finding it very difficult, but I get zero feedback: we don't have any tests and I was not able to make friends so I don't know how others are finding it. I'm starting to think that I might be an idiot.   How can I build some confidence in my abilities at the same time as I rush and struggle to keep up with the massive rate of new material I need to learn? How might I get myself some reassurance that I can actually pass, or stop worrying about this so much? (I feel terribly guilty using so much money from my family to be here) It's possible that I don't really have a chance at all but knowing that would be fine. We want to have a plain empty sandbox - one without any configuration metadata or data from Production.  Is it possible to have such a sandbox?  Please advise. Why not use "which"? What to use then? What is the proof that the total number of subsets of a set is $2^n$? What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set). Flag abusive users I think the community does a great job policing questions/answers, etc. and the community has some good tools available to do this.  Between flagging questions with some reasons behind it and editing posts, closing them, and deleting them, the only thing that seems to be missing to me is the ability to flag a user.  Reasons to flag a user as abusive:   Duplicate User Abusive Behavior Offensive user name or profile User profile consists of spam Others? What is the difference between Fact and Truth? I'm curious about the difference between Fact and Truth. I was searching on the internet if I could find it. But still I'm confused about the exact meaning. I first read the forum discussion here  where an author has given two examples for each like below     A fact is a reality that cannot be logically disputed or rejected. If   I say "fire is hot," I don't care how great your reasoning skills are,   if you touch fire your skin will burn (and don't give me that "but   people can walk on hot coals!" bull. There's a difference between the   transfer of heat through conduction and training one's body to deal   with the agonizing pain of said conduction). Now when I say this, I am   not speaking a truth, I am speaking a fact. If you say "fire is not   hot," you are not lying, you are incorrect. Facts are concrete   realities that no amount of reasoning will change. When one   acknowledges a fact, they are doing just that. Facts are not   discovered, facts are not created, facts are simply acknowledged.      A truth on the other hand, is almost the opposite. Truths are those   things that are not simply acknowledged, but must be discovered, or   created. If I say "God exists," and I possess strong reasoning for the   affirmative of that statement, then God really does exist, that is a   reality. However, if another individual possesses strong reasoning for   the negative, and because of this reasoning they believe that God does   not exist, then that is also a reality. If we were to debate our   ideologies, and my reasoning appeared stronger than theirs, they may   choose to adopt my belief that God does exist. If they do, then the   existence of God is just as true as the nonexistence of God which they   believed a week ago. Truths, as opposed to fact, are much more fluid   and malleable than their empirical counterparts.   and followed by further discussion.  Then I found this . Article from above link says like below:     Facts are notes and lyrics on sheet music.  Truth is what the singer   gives to the listener when she’s brave enough to open up and sing from   her heart.   But still curious about the difference between both of them.  In our daily life, in general conversation, we generally use these both terms interchangeably. Then what is the difference? Are they synonym or have specific difference? Prove that if $n$ is a natural number and $(3n+1)$ &amp; $(4n+1)$ are both perfect squares, then $56$ will divide $n$.   Clearly we have to show that $7$ and $8$ both will divide $n$.  I considered first $3n+1=a^2$ and $4n+1=b^2$. $4n+1$ is a odd perfect square.  - so we have $4n+1\equiv 1\pmod{8}$; from this $2|n$ so $3n+1$ is a odd perfect square.  - so $3n+1\equiv 1\pmod{8}$ so $8|n$ but I can't show $7|n$. How do I show this?  Thanks for the help. I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$ \varepsilon vs. \epsilon I have two large external drives with about 2.5 TB of data. About half the storage is images and videos that I want to put on a new NAS drive/device. I don’t want to carry over junk, duplicates and non photo/video files.   I am hoping Lightroom can help scour the mounds of folder structures on these drives and copy images/videos the new drive that I want to catalogue and keep clean as a working platform for all images and video. Once completed I plan to format the original drives and copy the cleaned-up content back to them as part of a backup strategy.  Can Lightroom help? If so, how? If not, is there some other application for Mac OS X that I can use for help with this? One problem I had with my original attempt is I wasn't sure how to move images and rename folders to get the new drive contents to make sense, so I formatted it and am looking to try again.  (steps for this operation would be helpful) What is the name of this Ubuntu program? A couple month ago I installed a bunch of clipboard manager programs in Ubuntu, tried them out and then removed them. I must have missed one of these and now it keeps bug me in my daily usage. I need some help to identify the name of this program because the program itself provides very limited information.  This clipboard program has below characteristics: 1. It occupies hotkey Ctrl+;. When I press this hotkey, the clipboard history will pop out. It happens in bash, Chrome, Eclipse IDE, and etc, wherever it is text inputable. This is kind of annoying because it overrides my other hotkeys inside these applications. 2. Very laconic UI. Like below picture, with only a line "Select to paste", then listing out the clipboard history. There is no program information whatsoever. Right clicking on it and no menu are shown at all.      So far I have tried below ways to identify this program: 1. Browse system monitor for processes with names that could be correlated. 2. Browse through my installation log (dpkg.log) to find my installation history. Without luck, the log is lost. But I was not successful. Order of nested random effects in lme4 I just fit a model in lme4, and I'm wondering what the heck I fit...  I have individuals id, and each is measured pass/fail on items that can be described using two factors, f1 and f2. My theory says that id and f2 can interact. So I want to compare these models:  # f2 doesn't matter m1 &lt;- lmer(pass ~ f1 + (1|id), family="binomial")  # f2 is a fixed effect, doesn't interact with individuals m2 &lt;- lmer(pass ~ f1 + f2 + (1|id), family="binomial")  # f2 is a nested random effect, interacting with individuals m3 &lt;- lmer(pass ~ f1 + (1|f2/id), family="binomial")   First, am I right that there's a random effect of each level of f2, and each individual's level of f2 is partially pooled, depending on the number of observations for that individual?  Second, does it make sense to fit f2 as a fixed effect with random individuals nested within each level of f2? How does one write that?  Third, what's this, with f2 and id reversed in the formula? I typed it by accident, but now I want to understand it.  m3weird &lt;- lmer(pass ~ f1 + (1|id/f2), family="binomial") In what cases do I use malloc and/or new? Is it possible to recursively instance object particles? I plugged in my external harddrive (which was formatted on my Mac into HFS+ journaled) to my Ubuntu desktop 9.04 64bit.  I am not able to get the drive to mount with write capability, how do I do that?  Right now all I'm getting is read access, I tried  sudo mount -t hfsplus /dev/sdf2 /media/"Portable HD"   but that still gave me only read access...  ideas?? Is there a way in photoshop to convert k values to alpha values? Basically, I would like to make it so that the lighter a pixel is (the more white it has) the more transparent it becomes. Black pixels would be opaque, white pixels would be transparent.  I currently use the slider in blending options on the first tab to get rid of the white and make it transparent but find it works slightly better if I use the magic wand tool with delete.  I figure that there is some blending option out there but there are so many and I'm not sure which to use to achieve my goal.  (Using Photoshop CS6)