Prove that the set  $\{\cos(n) \mid n \in \mathbb{N}\}$ is dense in $[-1,1]$.
Show that $\{\cos\ n:n \in \mathbb{N}\}$ is dense in $[-1,1]$ by using the fact below:      Suppose $x$ is irrational. Then there exists $p_n,q_n \in \mathbb{Z}$   such that $\bigg|x -\frac{p_n}{q_n}\bigg| &lt; \frac{1}{q_n^2}$   I have no idea on how to apply the fact above to show the set is dense in $[-1,1]$. Can anyone help me?
Copying and pasting text between applications without style When pasting text, some applications (Chrome, Microsoft Office, etc.) will do what they can to re-render copied text with style, and some of those applications do not appear to offer an option to paste without style.  This results in spending lots of time removing formatting (where it is even possible) and then cleaning up the mess left behind, just to get clean text.  I want to prevent text styling from attaching itself to text that I copy and paste between applications. Is there an OS-level way to do this, i.e. one that can apply to all applications? Failing that, is there a per-application way to tell Chrome, Safari, Word etc. not to copy and/or paste with style? How to repair a corrupted Excel file I'm trying to restore some files for a friend who messed up a USB stick by pulling it out of the computer too soon. By searching invisible partitions I was able to restore the files. All files are intact, except 1 excel file that seems to be corrupted.  I've googled about ways to repair an Excel file, because the default repair options in Excel 2003-2010 do not work for this file. I've came across a numerous  third-party apps, of which the  one seemes to be able to do the job: Recovery Toolbox for Excel.  Of-course, this and all other apps I found require a purchase to actually repair the file. It seems a bit silly to pay 30 dollars for a one time thing, and something that is not for myself. So, I would like to keep that as a last-resort option.  Is there anything I can try to fix this? I've attached a screenshot of how the file looks when I open it on my OSX when converted to a HTML file (to view the code), for  the option that it could be a character set problem. As far as I know, the variance inflation factor is not computed with pseudo-$R^{2}$ or generalized $R^{2}$ in binary outcome models (e.g. logistic regression).  Are there other measures of multi-colinearity than VIF appropriate to such models?  Why ought or ought not we think about multi-colinearity in such models? Why was this scene from the trailer not in the film? I had seen all the trailers of  before its release. In one of those trailers Uncle Ben (Martin Sheen) says to Peter,       Do you have any idea what you really are?   This seems to imply that Uncle Ben knows about Peter's identity. This would have been a huge spoiler. But when I saw the movie, there was no such scene.   Do the filmmakers discuss such spoilers with some of the audiences and see if they like it and then change it accordingly? Or was this scene cut for other reasons? How can I make sure that a certain OLEDB driver is installed when I start my application? I use ADO from Delphi and would like to display a descriptive error message if the driver is missing. The error that's returned from ADO isn't always that user-friendly.  There are probably a nice little function that returns all installed drivers but I haven't found it. How to disable sticky corners in Windows 10 It seems Microsoft has tried to solve this problem .  In Windows 7, there is no boundary between monitors and the mouse can move freely across the top of the screen between my 3 monitors, provided I had set their heights equal in display settings. Windows 8 introduced sticky corners. See .  Windows 10 got sticky corners as well. There are a few pixels at the top corners of each display where the mouse cannot cross over onto the other display. One must move the cursor down to avoid this region in order to get to the next display.  The image shows roughly the region where mouse movement is not permitted in Windows 10, but is allowed in Windows 7.    Personally, I had no problem with unrestricted mouse movement across the top of my screens - I got used to "aiming" for the x, and the convenience of unrestricted cursor movement. Like all the people who wanted to disable it in W8, I'm wondering if there is a way to disable it in W10.  Edit to address possible duplicate:  Although the problem is identical to the one in , solutions to solve the problem in W8 involving editing the registry key MouseCornerClipLength do not work in W10, since that registry key is not present in W10. Also adding that key and setting the value doesn't work. I searched the entire registry and couldn't find it in another location. No other W10 keys in the node referenced in the W8 solution are obvious replacements.  Edit to address possible solutions in comments  harrymc's  and  that worked for for Windows 8.1 do not work in Windows 10. Add AD users to sudo group not working | CentOS 6.7 Can I keep the Rueful Axe and get Clavicus Vile's Mask? Is there a way to keep the Rueful axe and get Clavicus's Mask?  I thought you could kill Barbas after receiving the mask, but he is a statue now.  Did I miss my chance? How to construct semi direct products between Q and H There is a semi direct product if you go $\theta: Q \rightarrow Aut(H)$.  $H = C_{17}, Q = C_2$.  $Aut(H)\cong C_{16}$.  From here, how do I construct the semi direct products?   I said, in $C_{16}$, there are two elements of order that divide $2$, which are $1$ and $2$. Therefore there are two semi direct products. Also $17 \times 2 = 34$ tells me that in both semi direct products there will be $34$ elements.  How do I construct the semi direct products from here though? How do I properly install a system app given its .apk? I removed a system app (com.android.mms) and I have the .apk needed to restore it, however it won't install through the standard channels (running the .apk gives me "application not installed"). What's the proper way to install a system app's .apk? How can I include the cursor in the screenshots created when PrtScr is pressed? What is time, does it flow, and if so what defines its direction? Trouble downloading packages list due to a "Hash sum mismatch" error Mutineers force captain to record instructions to spaceship's computer but he leaves out "please" The crew mutinies on a spaceship whose captain controls it with verbal orders. They test whether they can control the ship with recordings of the captain's voice with an order to increase orbital speed (currently orbiting a planet). It works, so they force the captain to record all the possible orders they may need, and execute him. Only then do they realize that the ship obeys only orders couched in polite language, and only the test recording was like this, but the other recordings are not. They will die once the ship's faster orbit somehow causes it to run out of fuel, or something like that. I seem to remember the ship's &quot;computer&quot; following the captain around on robotic legs. I read the story circa 1980, but it might have been published much earlier. What do you do in Dungeon World when someone wants to perform an action they don't have a move for? How do we see? Where do the photons disappear? How to add a confirm dialog to a command button? I'd think this would work just fine, but it seems to prevent the action function from firing regardless of the confirm response  &lt;apex:commandButton value="Remove All" action="{!removeAll}" rerender="form"      status="LoadingStatus" onclick="return confirm('Are you sure?');"/&gt; I would like someone to explain me how the sense of a sentence changes with using "for me" or "to me":  Some examples:     1a.  She is a good friend to me.   1b.  She is a good friend for me.        2a.  How special you are to me.   2b.  How special you are for me.        3a.  It's an interesting topic to me.   3b.  It's an interesting topic for me. What is the best way to copy a folder and all subfolders and files using c# Geomagnetic reversal?