Subspace of a separable metric space is also separable
Prove that a subset of a separable set is itself separable
I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. I've used Windows for many years, using Dreamweaver as my editor as it had a file tree which is based on a project (via Local, FTP or SFTP).  I need a web development IDE with the following functionality:   syntax highlighting for PHP, HTML, CSS, JavaScript  file tree - Local, FTP, SFTP, Project-Based (e.g. via the methods of Local/FTP/SFTP) in other words, like Dreamweaver can open multiple files (tabbed)   Additionally, I wouldn't mind features like debugging, class listings, etc, as long as I can hide them away. Cannot update grub with parameters on live USB I have booted from a live USB ("Try Ubuntu"), that also has a persistent option set (I used  to create one) to do some tests for .  I'm trying to test  (like ) by doing this  sudo nano /etc/default/grub sudo update-grub   The problem is that that last command gives me this:  /usr/sbin/grub-probe: error: failed to get canonical path of /cow.   It looks like /cow is the file-system that is mounted on /, according to:  :~# df Filesystem     1K-blocks    Used Available Use% Mounted on /cow             4056896 2840204   1007284  74% / udev             1525912       4   1525908   1% /dev tmpfs             613768     844    612924   1% /run ....   Is there a way for me to run update-grub? Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$ Prove $$\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n! \qquad\text{and}\qquad \sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0,$$ where $m \in \{0, 1, 2, \cdots, n - 1\}$.  To be honest, I tried to use some combinatorial ideas but was not able to see interconnections. Also, I had an idea to prove it using induction but failed to do that either. Help will be much appreciated! I just got stuck and don't see any continuation of solution for this problem How can I tell if a laptop's video card can be upgraded? Is it true that $0.999999999\ldots=1$? I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? Can I set the start point of a bezier curve? Why am I getting different results from two servers that located in the same destination network? There's two servers with the same network IP, and the only difference between them is the last two numbers like: 37.211.15.247 and 37.211.166.178. The Hosting Provider claims that the subnet mask for both networks is 255.254.0.0  When I try the command Tracert (Traceroute) to test the ping and to see what hops does it take to reach the destination server, I get different ping times and different forward path results which is strange because they are all located in the same company host.  Is there something wrong? Shouldn’t it take the same path too?  I have added the the full IP addresses to both servers as requested by some commenters. Well the two servers aren’t “International” anyways, so as far as I know, the only thing that comes to mind is the possibility that the two are configured on different subnetworks just like what . Though it’s suspicious.  UPDATE: After doing so much research, I found out the Subnet of these two hosts. They are on a /15 (255.254.0.0) subnet mask. Unfortunately, this is what that hosting provider claimed to me which is not true, and the true Subnet they apparently use is a /17 (255.255.128.0) Subnet mask. It turns out that they were definitely on different Subnets. That's why I got different traceroutes when I tested these two IPs 37.211.15.247 and 37.211.166.178  because they are clearly on different Subnets. Thank you everyone for these good answers, especially  his answer helped me a lot. When I try to set the default combo for switching to different language layouts in my keyboard from Shift+Super+Space to Alt+Shift I can't.  I used to have this combo since Ubuntu 14 and even before that when I used to have Windows. Now for some stupid reason it does not allow (actually nothing happens its not like I get an error or the combo is used elsewhere). How could I bypass this bug? What happens when my aasimar Celestial warlock has overlapping class and racial features? If you are an Aasimar warlock whose otherworldly patron is The Celestial, you will know the light cantrip from both race and class. Additionally, once you reach 6th level in warlock you will have resistance to radiant damage from both race and class.  Is there any ruling on what happens when you have overlapping class and race features? Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement why __getitem__ cannot be classmethod? Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot! What is the sweet spot for experience gain? Is it more effective to fight the highest level enemies I can find, or am I penalized for being out of my depth? Are certain areas of the game better/best for experience gain/leveling? Are bosses better than areas with infinite minions? Would I get exp faster in a higher-level co-op game? What are the most time-effective techniques for rushing/boosting a character (either solo or co-op)? I need to calculate the time complexity of the following code:  for (i = 1; i &lt;= n; i++) {   for(j = 1; j &lt;= i; j++)   {    // Some code   } }   Is it O(n^2)? Cardinality of equivalence relations on N When is it correct to use "yourself" and "myself" (versus "you" and "me")? I'm confused by why people use the following:     It's up to yourself.   Rather than:     It's up to you.   Another example of this would be:     Please feel free to contact ourselves if you have any problems.   Rather than:     Please feel free to contact us if you have any problems.   Are both of these correct?  Is there any reason for using the former? Are locks unnecessary in multi-threaded Python code because of the GIL? If you are relying on an implementation of Python that has a Global Interpreter Lock (i.e. CPython) and writing multithreaded code, do you really need locks at all?  If the GIL doesn't allow multiple instructions to be executed in parallel, wouldn't shared data be unnecessary to protect?  sorry if this is a dumb question, but it is something I have always wondered about Python on multi-processor/core machines.    same thing would apply to any other language implementation that has a GIL. Convergence of Power Series with $z^{2^k}$ Zero-pad digits in string