Compute, for every real value $x$, the double limit:   $$\lim_{n \to \infty} \lim_{m \to \infty} \cos^m(n!2\pi x)$$  How do i go about solving the above problem? Note that this is a problem from a real analysis course, not calculus.  Thanks for looking at my question.
I stumbled upon this "relation" (is the name correct?):  $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&amp;x\text{ is rational}\\ 0,&amp;x\text{ is irrational}\end{cases} $$  How is it called and why is it so? I'm really not asking for a proof since I fear it would be too complicated for me to understand, but rather for an "intuition".
In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. Pause button started working for Chrome / Youtube, messing with audio player controls How to create a shortcut that executes an xdotool command to simulate a key press? I'm trying to simulate a media key press using a custom shortcut.  So far, I've been able to achieve the simulation I need using the command:  xdotool key XF86AudioPlay   It works perfectly, it pauses or starts the music player every time it is run.  The problem comes when trying to run it as a shortcut.  First I tried to run an alias with a custom shortcut, but it did not work.  As explained on :     The commands run by keyboard bindings are not parsed via a shell. Your best bet is to create a directory bin in your homedir.   Next I created the suggested script and named it simplay:  #!/bin/sh xdotool key XF86AudioPlay   It is located inside a bin folder which is part of the PATH, this works fine and can be run from any location.  When trying to create the Custom Shortcut, I was unable to get it to execute, the configuration is the following:  Name: test Command: simplay  Shortcut: Ctrl+Alt+R   After searching a bit more I found another way to execute the shortcut:  gnome-terminal -e simplay   I tried it on a terminal and noticed that when executing it it would rapidly open a new terminal window, execute the child process, pausing the music and immediately close the window.  I changed the command on the shortcut and when hitting it I noticed the same behavior opening a terminal and immediately being closed but this time the child process did not pause the music, which makes me think that the script has been called on all my previous attempts but it is somehow not being able to execute.  How can I solve this problem according to your knowledge? How can I make my AES encryption identical between Java and Objective-C (iPhone)? Strange Bittorrent Log On My Server Prove $A = (A \setminus B) \cup (A \cap B)$ Get path to application? How can a user determine the path to an application being showed in launcher?  This is meant for non-programmers or experts, so terminal commands are not really suited for them. Why: we need to make our users able to drag files/folders on our applications, a standard OS feature entirely missing in Ubuntu. Does someone knows if it exists some ppa packages source for the newest Ubuntu version 19.10 i had problem with that when i upgraded from Ubuntu 19.04 to 19.10 i didn't work? Number of suggested edits seems to include own edits On the new History beta site I have the privilege to approve tag edits already but my own edits still go into the review queue. And the review dashboard displays a different number of pending edits depending on where I am: on  I see one pending edit, on  I see none. I am pretty sure that the pending edit being displayed is my own - I edited a tag wiki and that edit hasn't been approved yet. It shouldn't be counted no matter where I am. How can I make Windows Paint save by default as JPEG format? Every time I want to save an image I've pasted into the  application, it chooses a BMP/DIB format (which is the worst for saving things from the clipboard).  How can I hack Paint into defaulting to the JPEG format when saving images?  I am using Windows XP SP3 and Paint 5.1 at the moment. I hope though that any hacks might be generic and I'd be able to use them across all my Windows machines. C programming: How to program for Unicode? What prerequisites are needed to do strict Unicode programming?  Does this imply that my code should not use char types anywhere and that functions need to be used that can deal with wint_t and wchar_t?  And what is the role played by multibyte character sequences in this scenario? If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition  $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$  then is $f$ constant function? How to programmatically create a WPF window in a WinForm application Is there a fast way (without looping over all edges/faces/verts) using ops.mesh or bmesh to get the lists of indices associated with mesh islands as shown below? These islands are part of one mesh but are disjoint from each other.    I know how to select the boundaries by running the operator Select boundary Loop or calling bpy.ops.mesh.region_to_loop() but can't find any API calls to return the vert/edge/face indices associated with the individual islands. Endow $S^3$ with its standard smooth structures from the stereographic projections, i.e. $$\chi_1:S^3 \setminus \{(0,0,0,1) \} \to \mathbb{R}^3, \chi(x,y,z,t) = \frac{1}{1-t}(x,y,z) $$ and $$\chi_2: S^3 \setminus \{(0,0,0,-1) \}   \to \mathbb{R}^3, \chi_2(x,y,z,t) = \frac{1}{1+t}(x,y,z).$$   Do the same for $S^2$, call the charts $\eta_1, \eta_2$ (in the same order as the ones for $S^3$). Now, consider the complex projective space $\mathbb{C} \mathbb{P}^1$ endowed with the smooth structure given by the charts $\{(U_0, \varphi_0), (U_1, \varphi_1)\},$ $$U_0 = \{[z_0:z_1] \ \mid \ z_0 \neq 0 \} \text{ and } U_1 = \{[z_0: z_1] \ \mid \ z_1 \neq 0 \},$$ where $[x:y]$ represents the complex line through the point $(x,y)$ and $0$ in $\mathbb{C}^2$, and $$\varphi_0([z_0: z_1]) = \frac{z_1}{z_0} \text{ and } \varphi_1([z_0: z_1]) = \frac{z_0}{z_1}. $$   Now define the Hopf maps: $$h: S^3 \to S^2, h(x,y,z,t) = (x^2+y^2-z^2-t^2, 2(yz-xt), 2(xz+yt)) $$ and $$H:S^3 \to \mathbb{C} \mathbb{P}^1, H(x,y,z,t) = [(x+iy):(z+it)]. $$  Is there an easier way to prove that $h$ is a well-defined smooth submersion and that $H$ is a smooth submersion (with respect to the respective smooth structures)? By this I mean we can of course compute $\eta_i \circ h \circ \chi_j^{-1}, \forall i,j \in \{1,2 \}$ and $\varphi_k \circ H \circ \chi_l^{-1}, \forall k,l \in \{1,2 \}$ and prove that they are smooth (in the classical real-analysis sense) and that they are submersions, but this is a very tedious process. So, are there easier ways to do this? Will this maybe be easier if we switch the smooth structure of the unit spheres from stereographic projections to, say, normal projections (though we increase the number of combinations of charts to verify from 4 to 48), since they induce the same smooth structure? Are nuclear weapons simply non-existent in the Star Wars universe? The resistance is attacks the Starkiller Base with what appears to be pretty small time ammunition even by modern earth missiles, which requires a long running sequence of repeated flybys.  The objective, IIRC was to destroy a critical piece of machinery on the surface which would then destabilize the entire planet/weapon (feel free to correct in edit if I remember incorrectly).  OK cool, but then why the force wouldn't they just drop a nuclear bomb on it?  Have nuclear weapons been used any other time in Star Wars (including EU), or, like smart phones, have they just not been invented in that world? $x,y,z \in \mathbb R$ My attempt: Notice that if $x^2 + y^2 + z^2 \geq xyz$ and $x+y+z \geq xyz$. So : $$x^2 + y^2 + z^2 \geq x+y+z $$ But this is actually not always true, take the case when $ 0&lt;x,y,z &lt;1$ And you will see that this is not working. Maybe I’m wrong. Self-references in object literals / initializers Randomly place and scale objects? show all my unanswered questions from SE Any way how to get    my all un-answered questions  from across all SE sites?