$Prove: \overline{B(a,r)}$ is a closed set.   We will prove that $\overline{B(a,r)}^{c}$ is an open set.  let there be $x\in\overline{B(a,r)}^{c}\rightarrow ||x-a|&gt;r\rightarrow \exists \delta&gt;0: ||x-a||&gt;r+\delta$  How from here I can conclude that there is an open ball contained in $\overline{B(a,r)}^{c}$ and to finish the proof?
Prove that a closed ball in a metric space is a closed set   My attempt: Suppose $D(x_0, r)$ is a closed ball. We show that $X \setminus D $ is open. In other words, we need to find an open ball contained in $X \setminus D$.   Pick $$t \in X-D \implies d(t,x_0) &gt; r \implies d(t,x_0) - r &gt; 0 $$ Let $B(y, r_1)$ be an open ball, and pick $z \in B(y,r_1)$. Then, we must have $d(y,z) &lt; r_1 $. We need to choose $r_1$ so that $d(z,x_0) &gt; r$. Notice by the triangle inequality  $$ d(x_0,t) \leq d(x_0,z) + d(z,t) \implies d(z,x_0) \geq d(x_0,t) - d(z,t) &gt; d(x_0,t) - r_1.$$  Notice, if we pick $r_1 = d(t,x_0)-r$ then we are done.  Is this correct?
Is there a particular reason that using clone = obj *1 is avoided when "cloning" objects? I just got an Android tablet and when I try to install an app on the Play store it says:      error retrieving information from server [RPC:S-3].   I've tried clearing the data for the Play Store app &amp; turning it off &amp; back on but it didn't work. I have some thoughts on this. First, I want to say I am no expert on cryptography, I just know some stuff, and I took a cryptography class in University. I am very interested in this topic.  I understand that many security experts out there might get angry just by the discussion of these things, but I hope you can spare me :)  My Questions:  Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to 2^2048 ?  I understand this would take an immense amount of time to do, but it would only be done once.  Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  When I consider who might break such a code, I am primarily thinking about NSA and other Governments that might have spent years in research and have the resources and interest in doing so.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index ? How long would that take in permutations, and perhaps with the fastest computer in the world ?  I hope we can have a nice chat about this. We are just discussion theory here. I hope you can offer me some insight on this topic.  Thanks! Java FileInputStream ObjectInputStream reaches end of file EOF I came to this conclusion: the primes that can't be written as the sum of two squares (lets use$p\prime$ to denote) can be written as ${p}\prime \equiv 3 \;(\bmod\; 4)$.  I need a simple proof to show this and am stuck. I would appreciate guidance! Installing Steam on Debian I have ran into some issues when it went down to install Steam on Debian. The console as well as Apper comes up with missing dependencies  and the package isn't going to install. Convert a preprocessor token to a string I'm looking for a way to convert a preprocessor token to a string.  Specifically, I've somewhere got:  #define MAX_LEN 16   and I want to use it to prevent buffer overrun:  char val[MAX_LEN+1]; // room for \0 sscanf(buf, "%"MAX_LEN"s", val);   I'm open to other ways to accomplish the same thing, but standard library only. Are actions and bonus actions interchangeable? How do I make Units efficiently? I just blew almost all my money on a really cool lookin' ship. How do I regain solvency without spending a lot of time fiddling around in my inventory?  At the moment I've found a sequence of steps that'll turn 50 carbon and 100 Plutonium into ~5000 Units (by creating Suspension Fluid and then turning it into Electron Vapor). Are there better recipes or sequences of recipes I should look for? Or should I just drop the whole thing and go mine a particular mineral? Why is removing instationarities a good thing when trying to forecast a time series? What happens when I connect a 2 amp charger instead of the 0.7 amp charger provided? Possible Duplicate:             I have a phone with 0.7 amp charger. I also have a Galaxy Nexus with a 2 amp charger.  If I connect the 2 amp charger into my phone, will my phone blow up? I have just discovered that my local bank, Nordea, has decided to remove the only good thing about their entire operation: the so-called "simplified login" feature on their website, which allows (soon allowed) the customer to log in to a kind of very limited "read-only" mode which basically just allowed you to see your current balance, the last few transactions (limited descriptions) and see whether or not there are any new "customer support messages", but not read those messages, not make any kind of payments, edit accounts, etc.  Now, they will only allow "full" logins, which 100% prevents any kind of automation and requires you to sit with a hardware device and enter numbers into it and then into the webpage before it allows you to see as much as your account balance. Other than being very obnoxious for every customer, it also blocks me and other "advanced" users out from our custom-programmed bots which have always regularly logged in with that "simplified" login method just to regexp out the current balance/new messages and then notify us via whatever mechanism we use privately.  I have previously asked them if they have any kind of JSON/RSS API of any kind, but no. They don't even have as much as e-mail notifications! Nothing except their spyware "smartphone app" which I would never install, nor would I ever allow such a surveillance device to track me 24/7 in the first place. Only by being a registered corporation with special, way more costly account types do you get any kind of "API" access from them, even to do this most basic imaginable task.  Before you tell me to switch banks, the others are at least just as bad (even worse from what I can tell, and for a longer time), and the act of switching banks is practically impossible for me personally.  What I'm asking is if maybe somebody knows of some sort of "hidden standard" or something which all banks in the world support, but don't have documented, for people like me to automate basic things such as knowing my account balance. Needless to say, I have looked for this countless times on their website, asked them about it, and asked other people about it. Nobody seems to know anything, or, if they do, won't tell me.  I know that there are a bunch of "Nordea-supporting" PHP libraries and whatnot on Github, but they are all ancient and even if they were up to date, they were most likely using the same method as I have been using: automating the login process through their "simplified login" mechanism, which is "soon" to be gone. (Probably at new year; they didn't even mention any dates or time frames.)  And yes, I have a Bitcoin wallet and have that fully automated. The problem is that virtually nobody else in the world cares about my Bitcoin, so "switching to Bitcoin" remains a pipe dream which I would be ready to do, but the world clearly isn't. get all page IDs from wp_list_pages How can I enable multiple users login at a time in remote system? Recently we bought dedicated server with Windows Server 2012 Standard.It's allowing 2 people at a time.So I want to allow more people at a time.  How can I do this? The following exercise comes from Fulton's Algebraic Curves:     Let $k$ be an infinite field and $V = \{(t,t^2,t^3) \, ; \, t \in k \} \subset \mathbb{A}^3(k)$. Show that $V$ is algebraic and compute $I(V)$ by giving a finite set of generators.   Now, showing $V$ is algebraic is easy by noting that $V = V(X^2-Y, X^3-Z)$, but I can't find a way to describe $I(V)$ explicitly. Obviously $(X^2-Y)(X^3-Z) \in V$, but how can I find the actual generators of this ideal? Limit of a function with powers (L'Hopital doesnt work) I have a little problem with limit of this function:  $\lim_{x \to \infty} x^2(2017^{\frac{1}{x}} - 2017^{\frac{1}{x+1}})$  I have tried de L'Hopital rule twice, but it doesn't work. Now I have no idea how to do it. I'm new to TDD and xUnit so I want to test my method that looks something like:   List&lt;T&gt; DeleteElements&lt;T&gt;(this List&lt;T&gt; a, List&lt;T&gt; b);   Is there any Assert method that I can use ? I think something like this would be nice  List&lt;int&gt; values = new List&lt;int&gt;() { 1, 2, 3 }; List&lt;int&gt; expected = new List&lt;int&gt;() { 1 }; List&lt;int&gt; actual = values.DeleteElements(new List&lt;int&gt;() { 2, 3 });  Assert.Exact(expected, actual);   Is there something like this ? So, I'm working with OpenGL (I'm not exactly sure of the version), and I want to do an old-style lighting setup by essentially drawing a black rectangle over the screen, and drawing white circles over the areas that should be lit. I already have the rectangle and the circles, but I can't figure out how to make it so that the white circles are transparent when applied to the black areas. Any ideas? I tried to use blending modes, but it still evades me...  EDIT: Here's the code that I'm using (Python).  view_buf = Buffer(GL_INT, 4) glGetIntegerv(GL_VIEWPORT, view_buf) view = view_buf.to_list() if hasattr(view_buf, "to_list") else view_buf.list  glDisable(GL_DEPTH_TEST)  glDisable(GL_LIGHTING)  glEnable(GL_BLEND)  if logic.lights['smooth']:     glEnable(GL_LINE_SMOOTH)  # # Make sure we're using smooth shading instead of flat # glShadeModel(GL_SMOOTH)  # Setup the matrices  glMatrixMode(GL_PROJECTION) glPushMatrix() glLoadIdentity() gluOrtho2D(0, view[2], 0, view[3]) glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity()  #glAlphaFunc(GL_GREATER, 0.5)  glColor4f(0.0, 0, 0, 0.75)  glBegin(GL_QUADS);                              # Draw Black screen glVertex2d(0.0, 0.0); glVertex2d(view[2], 0.0); glVertex2d(view[2], view[3]); glVertex2d(0.0, view[3]); glEnd()  glColor4f(1, 1, 1, 1)  degtorad = 3.14159/180.0   glBlendFunc(GL_ZERO, GL_SRC_ALPHA)  def DrawCircle(x, y, radius, fill = 1, step = 11):      glVertex2f(x, y)      if fill:         glBegin(GL_TRIANGLE_FAN)     else:         glBegin(GL_LINE_LOOP)      for a in range(0, 360, step):         angle = degtorad * a         glVertex2f(x + (sin(angle) * radius), y + (cos(angle) * radius))      glEnd()   #for layer in range(len(logic.lights['layers'])): #   size =  for light in logic.lights['points']:     DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], step = light[2])     if logic.lights['smooth']:         DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], 0, light[2])  # Reset the state glPopMatrix() glMatrixMode(GL_PROJECTION) glPopMatrix() : So how many named special cases of each modal verb are there, in toto? There's   1. subjunctive should, and   2. mandative should, and   3. , at least;   and I s[up]pose 4. Should we? is suggestive should, and   5. Shouldn't we? is tag **should**, and I can think of a lot more.    I ask about only operative uses of this modal verb including literary and formal usage,  but please omit anything obsolete. I'm trying to classify them and learn the formal terms describing each. Are there any apt others in addition to the ones above? (I added the enumeration below)  1. What's 'subjunctive should'?  Whenever should is conjugated in the subjunctive mood? 5. What's a tag **should** ? 6. Where does  should/would  feature? What's it called? This should is detailed in   or  or . What visa does my wife need to travel to the UK with me, if I'm going there on a business visa? I am resident of India and just got UK Business visa ( 6 months, multiple entry) .. I am going for a week on business trip. I want to take my wife along with me and at the end of trip.. I want to extend my trip for 3 days and we want to spend some time on leisure.   What visa I should apply for my wife.. visitor visa? should she mention in the visa form that she is going along with me and staying with me while I am on work and later we are spending time on leisure.   While filling my business visa form, there was a field asking if anyone else is going with me.. I selected no.. as there were no plans.  Also is it ok to spend time on leisure on UK Business visa