Given a metric space $X$ and $E \subseteq X$. How can I show that $x \in \overline{E}$ if and only if there exists a sequence $x_n \in E$ such that $\lim_{n\to\infty} x_n=x$.
Say I have a metric space $(X, d)$ and a set $A\in X$.  I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help?
Math mode macro in TeXstudio I want to set up a macro in TeXstudio for math mode which displays $*$ and places the cursor at position *. So far I have tried using $%|$ but upon triggering, this macro does not allow text insertion at the placeholder.  EDIT: The method above works fine if the macro shortcut is set to the default Shift+F1. However, changing the shortcut to another combination, as I would like to do, seems to be the cause of the problem. How to show zero before decimal point in bc? echo "scale=3;1/8" | bc   shows .125 on the screen. How to show 0.125 if the output result is less than one? Better Information On Defense Damage Clash of Clans The way that Clash of Clans chooses to convey information about how much damage defense buildings output is bad. The current method of telling the user that a mortar does X dmg a second is pretty worthless since they fire much slower than once a second.  Where can I find a break down / what is the damage break down for how much damage these buildings do per shot rather than per second?  Sure, information on each level of the buildings would be nice but if that is too much I think simply providing the data for level one of each building would also work. I would like to know the model of my graphics card. I think it may be an ATI, but I want to be sure!   I have Ubuntu 11.10 (32 bit) and an Asus A6 VA laptop. Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count. I have a laptop with win7 installed. I have now made a 60gb partition which I want to install ubuntu into. The question I have, before I do the installation, is how large each of the root, swap, and home partition should be? I have read some place that root could be as small as 8GB, but isn't that too small? Since I guess beside ubuntu all the softwares installed will reside there as well? And I think I'm going to set my swap to be 2GB large.  My main concern is how large should the root partition should be. I'm mainly going to use ubuntu for programming and browse the web. Why doesn't 'find' show this file? Using find with grep, one can locate files that match a pattern:  # find | grep error ./solr-modifiedSolr4/SolrPhpClient/phpdocs/errors.html ./error_log ./includes/classes/error_log   However, using find alone the first file is not found:  # find . -name error* ./error_log ./includes/classes/error_log   Why doesn't find locate the errors.html file when not used with grep? How is find used to show this file as well? This is by far the weirdest problem I've come across,  My mac can perfectly access internet through safari web browser, but when I tried to do   ping www.google.com   through the terminal, it's requests is being timed out.  None of the other apps are able to reach internet either. Dropbox, Skype, Spotify all showing offline, except the browser itself  I'm guessing something bad with the DNS, but spend 2 hours trying to fix this with no luck Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and . I want my app automatically know user phone number by accessing their SIM Card I am traveling from Chennai to Atlanta. M first flight is to Frankfurt reaches at 8 30 AM. My second flight is scheduled at 10 45 AM.   Will the two hours transit time enough for boarding my second flight.  What should be my do's and dont's?  Update  Its more than enough. I was able to make it without any tension. What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? Resistance between two points in an infinite metal sphere/cube Let's imagine that we have a tridimensional metal object of infinite size, and decide to calculate the resistance between two arbitrary points. How would we go about doing this?  I have thought of two possibilities for calculating this, the first one being an adaptation of the conductivity (S/m) using the Pythagorean theorem to find a distance and isolating the resistance from there, treating it like we would treat an electrolytic liquid.  The second possibility I have been thinking about would be a 3-dimentional adaptation of the infinite-resistors problem (which I have only seen in 2d).  Would one of these methods result in giving a decent approximation, or would another method give better results? Correct position of "only" Which is grammatically correct?  I can only do so much in this time.  or  I can do only so much in this time. What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? Differences between astronomy, astrophysics and cosmology? To make a JavaScript class with a public method I'd do something like:  function Restaurant() {}  Restaurant.prototype.buy_food = function(){    // something here }  Restaurant.prototype.use_restroom = function(){    // something here }   That way users of my class can:  var restaurant = new Restaurant(); restaurant.buy_food(); restaurant.use_restroom();   How do I create a private method that can be called by the buy_food and use_restroom methods but not externally by users of the class?  In other words, I want my method implementation to be able to do:  Restaurant.prototype.use_restroom = function() {    this.private_stuff(); }   But this shouldn't work:  var r = new Restaurant(); r.private_stuff();   How do I define private_stuff as a private method so both of these hold true?  I've read  a few times but it doesn't seem like "private" methods can be called by public methods and "privileged" methods can be called externally. I need to find a reg ex that only allows alphanumeric. So far, everyone I try only works if the string is alphanumeric, meaning contains both a letter and a number. I just want one what would allow either and not require both. Why is cold fusion considered bogus? Cold fusion is being mentioned a lot lately because of some new setup that apparently works. This is an unverified claim.  See for example:         ( of that last link in the Wayback Machine, given frequent  from that page.)   While we should give the scientific community time to evaluate the set up and eventually replicate the results, there is undoubtedly some skepticism that cold fusion would work at all, because the claim is quite extraordinary.  In the past, after Fleischmann and Pons announced their cold fusion results, in perfectly good faith, they were proven wrong by subsequent experiments.  What are the experimental realities that make Fleischmann and Pons style cold fusions experiments easy to get wrong?  Would the same risks apply to this new set up?