Given a metric $d(x,y)$ on metric space X, here is a theorem: Let $\rho(x,y) =\sum_{n=1}^{\infty}2^{-n}\frac{d(x_n,y_n)}{1+d(x_n,y_n)}$. This metric induces the product topology on $X^{\omega}$ I intended to use the lemma below to prove it: $\mathcal B$ and $\mathcal B'$are the basis of $(X,\mathcal T)$ and$(X,\mathcal T')$. Then: $\mathcal{T \subset \mathcal T'}\leftrightarrow \forall x\in X,\forall B \in \mathcal B$ with $x\in B,\exists B'\in\mathcal B' s.t. x\in B'\subset B.$ This means I have to find a ball $B(x,\varepsilon) $ and a $\Pi^{\infty}_{n=1}U_n$(only finite $U_n \ne X)$, which I can't find. How to prove this?
Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$.  Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times X\rightarrow\mathbb{R}$ by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n[1+\rho_n(x_n,y_n)]}.$$  Show that $\rho$ is a metric on $X$ whose induced topology is equivalent to the product topology on $X$.  So basically what this problem is saying is that there's a canonical way to define a metric on the countable product of metric spaces.  I showed in a previous problem that the topology induced by $\rho_n$ is equivalent to that induced by $\frac{\rho_n(x_n,y_n)}{1+\rho_n(x_n,y_n)}$.  And thus we can go ahead and just assume that $\rho_n&lt; 1$ for all $n$ and replace our infinite series by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n}.$$  Now comes the interesting part: how should I go about showing that the product topology on $X$ and the topology induced by $\rho$ are equivalent?    The basis for the product topology given to me in my book's definition is that of cartesian products made up the $X_n$ except for finitely many which are $O_n$ for some open subset of $X_n$.  However I believe I was able to improve upon this and show that I could decompose these into a basis where the $O_n$ were all open balls induced by their respect $\rho_n$ metric.  For $\rho$ I'm using the basis of open balls that it induces, as I can see no other reasonable choice.    However I can't seem to match these two bases up.  There are many different points $\{x_n\}\in X$ which make my infinite series less than a certain value and there is so much freedom in which terms in the series I choose to reduce in size that it seems hopeless to try and fit an open ball induced by $\rho$ into any one of my basis elements for the product topology.    Is there a more appropriate strategy for proving these two topologies are equivalent?
How to boot Ubuntu persistent live USB on Android? I'm discovering the possibilities offered by persistent live USB using Ubuntu 19.10 and I was wondering if it would be possible to boot with Ubuntu on Android (i.e. using your computer capabilities through your smartphone) using those multi-plug USB sticks which have both USB3 and micro-USB2.  In other way: can I get some kind of a boot menu on an Android device in order to boot with a persistent live USB in order to access my Linux computer tools and setup from a wider range of devices? I want to debug a router that is giving strange DHCP settings, but I don't know how to check it without having to make my system take all the configurations. Isn't there a utility that returns all DHCP settings (like NTP, WINS, DNS, IP, router) whenever a client connects?  Preferably that is platform independent, and that won't require changing the client's configuration to trigger the DHCP. Is there a better alternative to ambiguous for a word with a wide range of meanings, difficult to find, because they become different in connection with person's opinion. Why do popular but not ideal questions keep being closed? I notice a really worrying trend of some really popular questions being closed and eventually deleted. Some of them became legends already, like this one  ? Yes, it does not fit the format ideally, but what is point of obeying the letter of the law to such extent that it becomes ridiculous? Ok, closing is fine if community leaders became so pedantic, but why go so far as to even delete it? At this point it looks like burning old pictures just because you don't like them.  Update: Summary  For now I heard only two clear arguments:   Those questions pollute search results.  Those questions don't play well with enforcing question and answer standards.   Both of that statements are easily verifiable and to my experience are just false. What Lagrangian counterterms might be needed in a $\phi^ 6$ theory in 3D? How to send array with CURL? Should I `urlencode` it? Dual boot Win 10 + Ubuntu SSD+HDD first of all thank you for answering.  I have a 120GB SSD and a 1TB HDD and I want to dual boot with Windows 10. I tried with this configuration:  SSD  / (ext4 20GB) /usr (ext4 15GB) /boot/efi (ext4 250MB)  HDD  /var (ext4 3GB) /tmp (ext4 8GB) /home (ext4 50GB) swap (none 24GB) (RAM 16GB)  I don't know if I am doing OK, as I tried to install it but installation gets stuck at "Saving installed packages..." and it doesn't continue. I wonder if it could be cause I didn't  partitioned as I had to. I don't find any post where this happens so I ask here.  Thanks! Is there a tool to get the lines in one file that are not in another? How to set a maximum execution time for a mysql query? I would like to set a maximum execution time for sql queries like set_time_limit() in php. How can I do ? #!/bin/bash - no such file or directory I recently updated my operating system from Ubuntu 13.04 to Ubuntu 13.10. Since then, I am facing difficulties compiling my TeX code in which the french babel package is called. Please note that all my sources were compiling very fine before the system upgrade. However, now, given the following header:   \usepackage[francais]{babel}  I receive the following error:  ! Package babel Error: Unknown option `francais'. Either you misspelled it   What should I do? Please note that if I remove this part of code (\usepackage[francais]{babel}), everything is compiling well. I also tried it with french, and I receive the same result. Thank you in advance for your help.  MINIMUM WORKING EXAMPLE : \documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[francais]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}  \begin{document} This is a MINIMUM WORKING EXAMPLE. \end{document}   ! Package babel Error: Unknow option `francais'. Either you misspelled it I have been running Ubuntu 12.04 for the past few weeks. I recently tried to download and install some files through the Update Manager. It located a few files, downloaded them, but then failed to install them. It gave the error:  installArchives() failed: (Reading database ...  (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error   I tried using apt-get upgrade but a similar error was produced:  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be upgraded:   icedtea-6-jre-cacao icedtea-6-jre-jamvm openjdk-6-jre openjdk-6-jre-headless   openjdk-6-jre-lib 5 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/34.6 MB of archives. After this operation, 1,024 B of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   Any help would be greatly appreciated. Thanks in advance.  Edit: Here's some extra information that may be useful. My original problem was that the Ubuntu Software Centre would not open. I found that the Update Manager and Synaptic Package Manager also failed to open. They all gave an error along the lines of:   "installArchives() failed: dpkg: error: reading package info file '/var/lib/dpkg/available': Input/output errorError in function:"   After much googling I seemed to fix that problem as the package managers would open again. When I then tried to download and install any updates from the Update Manager I got the error at the beginning of this post. Extract date from a variable in a different format Let me explain you the problem   $ date +%c -d "$d" Tue 31 Dec 2013 01:13:06 PM CET $ date +'Today is %F' -d "$d" Today is 2013-12-31   This solution corresponds to current date.  But I have one variable which stores date other than current date  $Prev_date="Wed Dec 25 06:35:02 EST 2013"    I am looking for solution to read this date as 2013-12-25 and store it in a variable.  I have tried this:  a=`date --date=$Prev_date '+%y/%m/d'` echo $a   It's giving this error:  date: illegal option -- date=Wed usage: date [-u] mmddHHMM[[cc]yy][.SS] date [-u] [+format] date -a [-]sss[.fff] Is there a good tool/program that allows me to clone a system drive to another, smaller hard drive?  I've bought a new 64GB SSD drive and would like to move my current Windows 7 Professional 64-bit installation to that new drive. My current drive is a 200GB SATA drive with one partition only. Of course there's enough free space to shrink my current partition to less than 64GB. MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! Suppose there are $2$ permutations given by $p$ and $q$. I need to check whether $p$ belongs to the group generated by $q$, and if so, it's representation in a power of $q$. In other words, I've been given $p$ and $q$, I need to check whether $\exists i \in \{ 0,...,|q|-1 \} \text{ such that } p = q^i$ I know that if $m = \gcd(i, |q|)$, then $|q^i| = \frac{|q|}{m}$. Now, say that such an $i$ exists. Then, $$ |p| = \frac{|q|}{\gcd(i, |q|)} \Rightarrow \gcd(i, |q|) = \frac{|q|}{|p|} $$ Thus, a necessary condition for such an $i$ to exist is: $|p|$ divides $|q|$ Say that this holds, and $\frac{|q|}{|p|} = r$, where r is an integer. Then we just have to solve $\gcd(i, |q|) = r$ for $ i \in \{ 0,...,|q|-1 \}$. Can this be done? If not manually, then by some efficient algorithm? (Note: Just looping over all values of $i$ is not efficient as $i$ may be very large) Edit: Another way I can think of solving this is by writing both $p$ and $q$ as product of disjoint cycles: $$ p = c_1c_2\ldots c_k \text{ and } q = d_1d_2\ldots d_l $$ And thus, as disjoint cycles commute: $$ p^i = c_1^ic_2^i\ldots c_k^i $$ How to proceed after this? Does the Black Panther suit give extra speed? So in Civil War, the Black Panther suit seems to give the suit holder some extra speed, because the guy was running as fast as Bucky and Captain America.  So does it give extra speed? If so, how does that work? Game Theory and Uniform Distribution question? In an Auction , two players are  bidding. Their bids will be a unknown fraction of their valuations. The valuations come from a uniform distribution $$[0,1] $$  If Player 2 bids $$ v/2 $$  and Player 1 bids $$b1&lt;1/2$$   What is the probability player 1 wins ? Clearly for player 1 to win,players 2 bid has to be less than player 1 bids. $$P(v/2 &lt; b1)$$  $$P(v &lt; 2b1)$$ I follow the question up to this stage. Now it says since its uniformly distributed the probability player 1 wins is $$2b1$$ Im confused how can you just get 2b1 from the inequality ?  Thanks in advance Is there a way for non-root processes to bind to "privileged" ports on Linux? It's very annoying to have this limitation on my development box, when there won't ever be any users other than me.  I'm aware of , but none of them do exactly what I want:    (The version in Debian testing, 1.0, only supports IPv4)  (the "nat" table is not yet implemented for ip6tables, the IPv6 version of iptables) sudo (Running as root is what I'm trying to avoid) SELinux (or similar). (This is just my dev box, I don't want to introduce a lot of extra complexity.)   Is there some simple sysctl variable to allow non-root processes to bind to "privileged" ports (ports less than 1024) on Linux, or am I just out of luck?  EDIT: In some cases, you can  to do this. "Being" or "to be"? Which is better structured?     "She loves to be herself instead of showing off"    or     "She loves being herself instead of showing off"   or     "She loves being and not appearing"