How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
Graph of a continuous function is closed
How to prevent different checksums for different typesets I wasn't aware of a difference, but a coworker says there is, although he can't back it up. What's the difference if any? Can I run PC steam games on a Mac? Configuring dpkg stops processing when calling an HTTP request to Adobe Flash Player I recently moved to Linux, and I am in love with the experience. Opinions apart, I am having problems when running sudo dpkg --configure -a  My computer stops processing the command here:  You may know Adobe Flash Player is no longer supported, and I believe that could be why the command is unable to load pepperflashplugin-nonfree. I have tried fixing the problem by checking older posts here, but none of them was able to fix this problem. What does “{} \;” mean in find (in Linux)? Sometimes I see a command like  find . -name * -exec ls -a {} \;   I was asked to execute this.  What does {} \; mean here? If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$. Give moderators the option to select a deletion reason when removing posts There are many times when I curse the autocorrection mechanism for changing what I typed, but overall, much of the corrections are welcome so I don't really want to disable it. I do want to have it do it's work but alert me when it makes each change so I don't have to proofread everything or focus with perfect attention to the screen to see the momentary red and/or blue underlining that alerts me to the machine override.  I would love a way to turn on an audible beep (customizable of course) so that I also get a warning each time my typing gets corrected (and only when autocorrection occurs). Simulating Key Press C# I don't know how to prove this example. Let $y(t):\mathbb{R}^+\to \mathbb{R}$ be a nonincreasing function. Assume that there is a constant $A&gt;0$ such that \begin{equation} \int_{s}^{+\infty}y(t)dt \leq Ay(s), \quad 0\leq s&lt;+\infty \end{equation} Then $y(t)\leq y(0)e^{1-\frac{t}{A}}$, for all $t&gt;0$. How do you scrape AJAX pages? Please advise how to scrape AJAX pages. Ubuntu Wifi drivers not working after getting update Space of probability measures "complete"? (In the other sense) Why does the following code does'nt work when i am using PDFLaTeX or just LaTeX ? I tried it in different editors as well.  This is the code   $\{f^'_n\}$   Surprisingly it works on Math.SE site How do I resize partitions using command line without using a GUI on a server? I only have access to the server via a terminal and I can't use graphical tools such as GParted! I want to create a new partition from a part of the root (about 768mb) for swap.  # df -h Filesystem      Size  Used Avail Use% Mounted on /dev/vda         20G  1.3G   18G   7% / udev             10M     0   10M   0% /dev tmpfs           199M  4.9M  194M   3% /run tmpfs           100M   12K  100M   1% /run/user tmpfs           5.0M  4.0K  5.0M   1% /run/lock Contradiction between law of conservation of energy and law of conservation of momentum? Yesterday, while studying a simple question, a rather strange doubt hit my mind:     Consider a ball of mass $m$ moving with velocity $v$ that collides with a wall in a perfectly elastic collision and rebounds back with the same velocity. Taking the direction of initial motion of the ball as positive, the change in momentum of the ball is $-2mv$ and the wall is clearly at rest.   So, in order to obey the law of conservation of momentum, the wall must gain a momentum of $2mv$.  Now, this itself confused me a lot. I then applied the law of conservation of kinetic energy to this situation, and clearly the magnitudes of the initial and final velocities of the ball are equal, so kinetic energy of the ball is conserved and hence the final kinetic energy of the wall must be zero in order to conserve energy.  Now, this contradicts with the law of conservation of momentum, which says that the wall must gain momentum of $2mv$.  One more point: If we are talking about a wall then it is an important fact, which must be considered, that the wall is fixed on earth and so the mass of the wall is equivalent to that of earth, say $M$. If the law of conservation of momentum is applied now then we get velocity of the wall, say $V=2mv/M$ which is no doubt an extremely small value. But, the law of conservation of energy strictly says that the velocity of the wall must be exactly zero, not even the smallest value is allowed (theoretically speaking).   Please clarify this and correct me anywhere if I'm wrong. I have tried my best to make it clear what I want to ask :). PHP: Right way to declare variable before use in loop I have a variable that is built in loop. Something like:  $str = ""; for($i = 0; $i &lt; 10; $i++) $str .= "something";   If $str = "" is ommitted, I get undefined variable notice, but I thought php auto-declare a variable the first time it sees undeclared one?  How do I do this right? Why multiple capacitors in parallel? Wondering why you would choose to use smaller capacitors in parallel vs one bigger one.  Here's an example:      Is this a component size and track width choice?  Thanks! The rules of Transfiguration: examples of making the inedible edible? Information from the official website of J.K.Rowling is generally considered canon. I remember at least one case when she 'changed her mind' after a particular rule has been  published on her website (what happens with a secret if a Secret Keeper dies). Nevertheless, the information from the website can generally be considered reliable.  A rather obscure piece of information is J.K.Rowling's Wizards' Ordinary Magic and Basic Aptitude Test () published on her website in 2007 and now accessible with a Time Turner.   In particular, we know that only one of the following options is correct (as the software allows you to choose only one option):     Which of the following is CORRECT?            a. Food can be conjured out of thin air.       b. Any object can be Transfigured into food.       c. Foodstuffs can be increased, Transfigured, summoned from a distance and magically cooked.       d. It is impossible to make the inedible, edible.   e. Food-related Charms are some of the simplest forms of magic.      W.O.M.B.A.T. Grade 3, 13 June 2007, question 8   The test appeared on J.K.Rowling's website before the release of Book 7, but we can almost safely assume that there was no time for J.K.Rowling to change her mind after compiling the test (though she must have submitted these questions weeks before they were eventually published).  As Hermione explained in Harry Potter and the Deathly Hallows:      Your mother can’t produce food out of thin air, no one can. Food is the first of the  to . It’s impossible to make good food out of nothing! You can Summon it if you know where it is, you can transform it, you can increase the quantity if you’ve already got some...   It means that the correct option is c, and therefore all other statements are false. So, apparently, you can make food out of something else, though not any object can be transfigured into food. Can you come up with any example of such transformation mentioned in the books? Integrating $\int \sin^n{x} \ dx$ I am working on trying to solve this problem:     Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Here are the steps that I follow in the example that I am reading:     $u = \sin^{n - 1}{x}$      $du = (n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx$      $v = -\cos{x}$      $dv = \sin{x} \ dx$            $\int \sin^n{x} \ dx =  \sin^{n - 1}{x} \cdot \sin{x} \ dx$      $\int \sin^n{x} \ dx = \underbrace{\sin^{n - 1}{x}}_{u} \cdot \underbrace{-\cos{x}}_{v} - \int \underbrace{-\cos{x}}_{v} \cdot \underbrace{(n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx}_{du}$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \cos^{2}{x} \ dx$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$   Here is where I get lost. How did we go from $\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$ to $\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$? Even more specifically, where did $\sin^{n}{x}$ come from?     $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$   I get this part.     $n\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx$      $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Could someone please explain what I am missing?  Thank you for your time.