Let $A=$ $\{$ $\sum_n a_n3^{-n}$ : $a_n$ $=$ $0,2$ $\}$.  Show that $A$ as a subset of $[0,1]$ is closed:  My idea consists of taking a point $x$ in the complement of $A$.  However, I'm not sure then how to proceed. Can every element of [0,1] be expressed as a sum of $b_n3^{-n}$?  Where  $b_n=1?$  would the complement of A be of that form? May I have hints?  Please use this definition of A rather than any alternative definition.
I have encountered the definition of Cantor set and its property. Cantor set is constructed by removing the middle third open set of each interval .So each time we get some union of closed sets. As $F_0=[0,1]$ $F_1=[0,1/3]\cup [2/3,1]$ $F_2=[0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]$ So on , That for $F_n$ we get the Union of $2^n$ closed interval . I know that finite union of closed interval is closed, but this is not true for arbitarly union. As I had a counterexample:  $\cup$ {{1/n}|for $n$ $\in N$} as this is not closed as $0$ is not in  that union. So what is the argument here to say that $F_n$ is closed for any $ n$ . As we are using this fact to prove the Cantor set to be closed as $F=\cap_{n\to \infty} F_n$ i.e arbitarly intersection of closed set is closed . Where am I misunderstanding? Any help will be appreciated.
Is there a nice solution to get a "presenter mode" for Latex presentations? Where did Harry's parents get all of their money from? how to output text to both screen and file inside a shell script? Currently I have a shell script which logs messages to a log file like this:  log_file="/some/dir/log_file.log" echo "some text" &gt;&gt; $log_file do_some_command echo "more text" &gt;&gt; $log_file do_other_command   When executing this script, there is no output to screen, and, since I'm connecting to the server via putty, I have to open another connection and do "tail -f log_file_path.log", because I can't terminate the running script and I want to see the output in real time.  Obviously, what I want is that the text messages are printed on screen and into file, but I'd like to do it in one line, not two lines, one of which has no redirection to file.  How to achieve this? pkexec command in a .desktop file I made a .desktop file for Androxyde's Flashtool (utility for Sony Xperia devices that I have to open with an executable file in its folder) that requires root privileges to use fastboot utilities. I used to make it work with gksu, but I'm on Ubuntu 15.04 and gksu is now old.  I tried to modify the exec line from Exec=gksu /home/natasha/FlashTool/FlashTool to Exec=pkexec /home/natasha/FlashTool/FlashTool Then, I read about pkexec doesn't allow to run X11 applications and so I override in this way:      The problem now is: It asks me the password but the Flashtool's GUI doesn't start. BUT if I execute that command in terminal, the program starts without problems. What can I do? I have a HP 14-AN001NA laptop. It came with Windows 10 installed on which the Wi-Fi worked fine anywhere in the house.   However I removed Windows 10 and installed Ubuntu 16.04 instead and since I have had terrible Wi-Fi anywhere aside from about three foot away from the router. I don't know if it's relevant but my Wi-Fi is not called wlan0 but wlo1 for some reason (I didn't change it).  When running lspci -knn | grep Net -A3     I get the following output:  05:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter [10ec:b723] DeviceName:   Subsystem: Hewlett-Packard Company RTL8723BE PCIe Wireless Network Adapter [103c:81c1] Kernel driver in use: rtl8723be Kernel modules: wl, rtl8723be ubuntu 18.04 LTS bluetooth [0cf3:3004] discovery not working How to loop over the lines of a file? Say I have this file:  hello world hello world   This program  #!/bin/bash  for i in $(cat $1); do     echo "tester: $i" done   outputs  tester: hello tester: world tester: hello tester: world   I'd like to have the for iterate over each line individually ignoring whitespaces though, i.e. the last two lines should be replaced by  tester: hello world   Using quotes for i in "$(cat $1)"; results in i being assigned the whole file at once. What should I change? Prove that $Au\cdot v = u\cdot A^Tv$ Let A be an $n$ x $n$ matrix and let $u$,$v$ $\in$ $\mathbb{R^n}$. Prove that $$Au\cdot v = u\cdot A^Tv$$  I tried using the fact that $A^Tu=A\cdot u$. However, I cannot seem to get to this result. Could anyone please help me out? Multiple forms in delphi Understanding MongoDB BSON Document size limit Pause button started working for Chrome / Youtube, messing with audio player controls How do I make a Theme from scratch for Unity? I'd like to make a theme from scratch for Unity.  What knowledge and information/skills are needed? Are there any tools or templates out there for something like this already? I don't have any Python experience so tools biased towards beginners are preferable. Eigenvalues of Householder matrix What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it  to me intuitively or with a simple proof? Can you decide to counterspell after another counterspell has failed? I zoomed in / out of one of my UI panels and I can't figure out how to zoom back out/ in to get it normal size.   How can I reset the zoom level of UI panels? What Canon lens would one use for Sartorialist-like street photography? You probably know Scott Schuman and his blog (). What kind of lens (that is not expensive) would you recommend for Canon 550D to achieve similar effect (in terms of DoF etc.)? I'm a newbie so I'm not sure how to properly interpret those photos' EXIF info. My guess is, that a similar effect can be achieved using .   PS:   Update: Thank you guys for all the answers, it's much more clear for me right now. I think I'll try the Canon 50mm f/1.4 lens. In  Deathly Hallows, when Harry, Ron, and Hermione are at No. 12, Grimmauld Place, Harry summons Kreacher from the kitchens at Hogwarts where the elf has been working on Harry’s orders given a year previously.  When the trio’s heist in the Ministry of Magic ends with Hermione inadvertently bringing Yaxley inside the perimeter of the Fidelius charm that protects Grimmauld Place, the trio leg it and dare not return there for fear that it’s now crawling with Death Eaters.  Before they leave for the Ministry in the morning, Kreacher is talking about making a stew or pot roast or something for dinner. After the trio go on the run and camp out in the tent, Harry thinks of poor Kreacher preparing a dinner they’ll never return to eat, but they dare not summon him lest a Death Eater tag along (like how Dobby ‘tagged along’ with Kreacher when Harry called him to have him follow Draco in Half-Blood Prince).  The next time we see Kreacher is during the Battle of Hogwarts where he leads the house-elves from the kitchen with battle-cries of “For Master! For Harry Potter!” (or something like that), so he’s clearly lost none of his new-found loyalty to Harry.  Harry is still Kreacher’s master throughout the book, and since he does not communicate at all with Kreacher after their escape from the doorstep, he cannot have ordered Kreacher to return to Hogwarts to work there again.  We know that house-elves can go against their masters’ wishes, but this usually entails them punishing themselves rather brutally, and Kreacher does not seem to have been constantly injuring himself for the better part of a year at Hogwarts.  So when and how does Kreacher leave his master’s house without orders and return to Hogwarts? How to convert a Collection to List? Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions. (Reference : Fraleigh, A first course in abstract algebra)     Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, $\forall a,b\in G$, is a group.   Because * is an associatie operation, we have to prove that: there exists $e\in G$ such that $e*a=a=a*e$ for any $a\in G$, and that for any $a\in G$ there exists $\alpha \in G$ such that $a*\alpha=e=\alpha *a$.  This is very confusing, first I tried to prove the identity exists. If we consider a=b, then $a*x=a=y*a$, and because of the hypothesis $x,y\in G$ always exists, now we have to show that $x=y$, but I just can't do this with out inverses, so I tried to prove that the inverses exists, however I have no idea how the identity looks like, I'm stuck.  And I tried to get unstuck, so I thought particular cases, first if $G$ had only one element, then what would happend if it had just two elements... but I remembered that we don't know if G is finite or not, and I think this makes things a little bit more complicated.  What can I do? I did the following proof and I was wondering if its valid. It feels wrong because I didn't actually test the case when purportedly $n$ is not prime, but please feel free to correct me.  Assume there exists $n$ such that $\phi(n) = 14$. Assume $n$ is prime. Then $\phi(n) = n-1$. Then here, n-1 = 14, so n = 15.  We know since Euler's totient function is multiplicative that $\phi(xy)$ = $\phi(x)\phi(y)$, so $\phi(15) = \phi(3)\phi(5)$, but alas $\phi(3)\phi(5) = 2\cdot4 = 8 \ne 14$. If $n$ is not prime a similar argument follows since we know then that $n$ must be composed of prime numbers by the prime factorization theorem.