I know most of the result regarding Continuity, Completeness. But I got stuck into a question which I found in a question paper. Let $f:X \longrightarrow Y$ Where $X,Y$ metric space.Then show that for any Cauchy sequence $\{x_n\}$ in $X$ if $\{f(x_n)\}$ is Cauchy in $Y$, then $f$ is continuous. Please help me. Thanks in advance.
Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.  Is $f$ continuous?  Let $f$ be continuous, is it uniformly continuous?
Can toddlers be taught to safely navigate stairs? What is the relation between normal extension and separable extension? What is the relation between normal extension and separable extension? Let F be the algebraic extension of K, if F is a separable extension of K,if and only if F is a normal extension of K? is this correct? Mounting Flashdrive - mounted to home directory I wanted to mount my flashdrive, and made the mistake of setting the mount path as /home/my_name. Now all of my documents, downloads, pictures, etc. are unavailable since my flashdrive has replaced my previous home folder.  When I go to terminal and type sudo umount /home/my_name I get the error message:   umount: /home/my_name: device is busy.         (In some cases useful info about processes that use          the device is found by lsof(8) or fuser(1))   Is there a way to fix this, where I can restore my old /home/my_name folder and then mount the flashdrive to a proper location? Windows 10 won't accept Firefox as the default browser Accepting an answer should automatically upvote it as well Is there ever a case where you would want to accept an answer, but not upvote it?  I can't think of a single situation where this would be the case.. So would it not make sense to have the "Accept" button upvote the answer too? I tried to manipulate a cube with the Boolean modifier. I tried to cut out an edge of the cube with a cylinder. Also, the shading result looks awful in FLAT SHADING mode, so I turned on SMOOTH SHADING.   Has anyone an idea how to achieve a better result. (I also cut out a Sphere first, this made the result less awful).   I used Blender Internal and rendered with smooth shading for smoothness.  Using flat shading looks okay, except the "round" cutouts :/  Face normals are okay.  First image is smooth shading.    EDIT: User Pisurquatre suggested the EdgeSplit modifier, works perfect to me (Maybe a little tweaking here and there): I'm trying to find a way of building a new power supply for my workshop, in order to keep it small and not so hot I will be using a spare xformer to step down mains to 24v and then a variable buck converter to adjust it (and CC mode, displays and blinky stuff).   I thought it would be a good idea to use a linear regulator on the output stage to smooth things out, but I can't seem to think of a control scheme that allows me to change the voltage using only one potentiometer. The buck stage is this topology but with a external PNP transistor.(It has already been prototyped, it works alright, I know the mc34063A is old)   The only idea I got that ""works"" is using the buck converter output as a reference for a linear regulator (IC or discrete parts), but using a SMPS output as reference for anything is a bad idea, due to the amount of noise and fluctuations.   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Is this correct design decision? how to shuffle characters of strings in bash? i have this script where it generates random password but it needs to be shuffled. I couldnt find a way to do it. Please Help me.  num=("0" "1" "2" "3" "4" "5" "7" "8" "9") special=("@" "#" "$" "%" "*" "-" "+") upper=(A B C D E F G H I J K L M N O P Q R S T U V W X Y Z) lower=(a b c d e f g h i j k l m n o p q r s t u v w x y z) lower2=${#lower[*]}   #${lower[$((RANDOM%lower2))]} upper2=${#upper[*]}   #${upper[$((RANDOM%upper2))]} num2=${#num[*]}     #${num[$((RANDOM%num2))]} special2=${#special[*]} #${special[$((RANDOM%special2))]} echo "${special[$((RANDOM%special2))]}${num[$((RANDOM%num2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}"   this is the output: @7nOyIaJ  how do I randomize this? I have a small utility that I use to download an MP3 file from a website on a schedule and then builds/updates a podcast XML file which I've added to iTunes.  The text processing that creates/updates the XML file is written in Python. However, I use wget inside a Windows .bat file to download the actual MP3 file. I would prefer to have the entire utility written in Python.  I struggled to find a way to actually download the file in Python, thus why I resorted to using wget.  So, how do I download the file using Python? Artinian affine $K$-algebra Add "upload media" button in meta box field I have a custom post which has a meta tag that needs to be using a file upload (for video files). I am wondering, what is the correct way of adding an "Upload" button that points to the WordPress media uploader and sets the url of the selected uploaded file to the text field that caused that associates the upload button.  I am not looking for code on creating the actual meta tag option, but for a way to actually add a Wordpress media upload button properly. There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? This is about the upcoming  Zack Snyder movie , where Ben Affleck is playing Batman. I just read an  that Christian Bale wanted to be Batman again. So I wanted to ask why wasn't he cast as Batman in this movie. I checked the Wikipedia entry but nothing suggests why Christian Bale was not there. Nolan is associated with the movie as an adviser and DC Entertainment is producing this one too. So sort of familiar turf, right? But how come Ben Affleck is the new Batman?  EDIT: I would like to add another interesting piece of information. Bale is aged 40 (born 1974) and Affleck is 42 (born 1970). So they have actually cast an older actor. I have a Canon 7D, which has a crop factor of 1.6x.  When using a Canon EF-S lens I get the focal length stated on the lens, when using a Canon EF lens I get the focal length times 1.6 when used on the APS-C body. At least that's what I thought.  I just compared two lenses:   Canon EF-S 15-85mm f3.5-5.6 Canon EF 70-200mm f4   .  The top photo is shot with the 15-85, the bottom photo with the 70-200 - both lenses are set to 70mm. Now why do they look the same?  I thought I would get 112-320mm (70mm*1.6 crop factor). What did I misunderstand about the crop factor calculation? Lightroom also displays the photos as having "70mm" and not 112mm. beamer - global transition shifts slide content Using beamer with the , I want to have a global transition effect (a fast fade) which I set using the following code:  \addtobeamertemplate{background canvas}{\transfade[duration=0.2]}{}   as found on  answer.  However, doing it messes up with my slides settings. It doesn't change every slides, only the \section ones that are slightly shifted to the right, adding a white line on the left, as seen here:   I tried using XeLaTeX and LuaLaTeX, as they are the only compilers I could use for my current font, but they both have the same issue.  It probably comes from the theme, however I have no clue why it would happen or how to fix it.  Edit :   \documentclass{beamer} \usetheme{m} \addtobeamertemplate{background canvas}{\transfade}{}  \begin{document} \section{test} \end{document}   Requires the mtheme to work Can I delete files or folders from /private/var/folders/? I have this folder /private/var/folders/bf/ with 3GB of what seems temporary files. Can I delete this folder? Why aren't logarithms defined for negative $x$? 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