Let $a,b\in\mathbb{R}$ and $C^1([a,b])$ the space of the function that are continuously differentiable on the interval $[a,b]$. For $f,g\in C^1([a,b])$, let $$ d_{\infty}(f,g):=\lVert f-g\rVert_{\infty}:=\sup_{x\in [a,b]}\lvert f(x)-g(x)\rvert. $$          Show that $C^1([a,b]),d_{\infty})$ is not complete.      My idea is to use  $$ C^1([a,b])\subset C([a,b]),  $$ i.e. to find a sequence $(f_n)$ in $C^1([a,b])$ that converges to some $f\in C([a,b])$ with respect to the supremums norm $\lVert\cdot\rVert_{\infty}$ but which is not continously differentiable.   Then, $(f_n)$ should be a Cauchy sequence in $C^1([a,b])$ that does not converge in $C^1([a,b])$ (otherwise $f$ would be continously differentiable).  Now, I do have to find such a sequence... and did not have an idea yet.  EDIT  Maybe on $[-1,1]$ consider $$ f_n(x)=\sqrt{x^2+\frac{1}{n}},~~f(x)=\sqrt{x^2}. $$  $f(x)$ is not differentiable in $x=0$, hence not continuously differentiable on $[-1,1]$. The functions $f_n$ are continously differentiable on $[-1,1]$ and uniformly converge to $f$.
The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be differentiable.  How can I illustate this statement using a counter example? Can I use $ f_{n}=\frac1 n \sin nx $ as a counter example?  Also, is $ C^1[0,1] $ is same as the space given by $ X=\{f\in C^1[0,1]:f(0)=0 \}$. Can I use the same example to show that this is not a Banach space?  More help is appreciated! Thanks!
On my Amazing 2GB EeePC Surf, I have no more space left!  This is quite normal, and it happens every now and then. Just delete a few large, unneeded files and everything should be good, right?  So I go and find a 2MB file in /tmp, and rm -rf it. Check free space again, and -- still nothing?!  I had to remove about 30MB of files before it registered some free space.  Now, my question is, what's up with this strange behaviour? Multiple submit buttons in an HTML form Let's say you create a wizard in an HTML form. One button goes back, and one goes forward. Since the back button appears first in the markup when you press Enter, it will use that button to submit the form.  Example:    &lt;form&gt;   &lt;!-- Put your cursor in this field and press Enter --&gt;   &lt;input type="text" name="field1" /&gt;    &lt;!-- This is the button that will submit --&gt;   &lt;input type="submit" name="prev" value="Previous Page" /&gt;    &lt;!-- But this is the button that I WANT to submit --&gt;   &lt;input type="submit" name="next" value="Next Page" /&gt; &lt;/form&gt;     I would like to get to decide which button is used to submit the form when a user presses Enter. That way, when you press Enter the wizard will move to the next page, not the previous. Do you have to use tabindex to do this? Notify instantly when a currently viewed post is deleted At the moment SE shows us current voting results, a notification for new answers and comments, etc., instantly. However, it doesn't show us when a post is deleted.  Please notify us also when a post is deleted. Then I don't have to type a (long) comment, only to notice the post got deleted in the meantime. glBufferData not working outside main function How can I prove that the Cartesian product of two countable sets is also countable? DML Statement in ‘after trigger’ and not in ‘before trigger’? Does the std::stack in the C++ STL expose any iterators of the underlying container or should I use that container directly? Manage SSH Users with MySQL backend English equivalent for the Persian idiom "send someone out in pursuit of black chickpeas" "To send somebody out in pursuit of some black " is a Persian idiom that implies 'to make or ask someone to run an errand so that you be able to have/ buy some time in order to deal with your private affairs in their absence', like having or continuing a (secret) conversation with someone else.   According to its etymology, since 'black garbanzos/chickpeas' were rarely available in the markets, when someone wanted to buy them, they had to search many stores for a long time, and most of time they couldn't find them at all.   Today, its meaning has changed somehow, and it doesn't necessarily mean that we ask someone to run an impossible errand, but it means we use that errand as an excuse to send them away for a while just to have time to do something in their absence.   Example scenario:   Suppose your classmate has come to your house and you are talking about some girly issues in the sitting room. Now you want to tell her a big secret about one of your classmates, but not in the presence of your younger sister who is sitting there too. So you ask her to go to the market and buy some snacks for you. In this way, you will have some time to continue your conversation until she is back. So you might say to your friend:     Well, I sent my sister out in pursuit of some black chickpeas.  I'm sure that she won't be back for another 15 minutes, so we can talk more freely now."   And your sister, before leaving the house, might sarcastically say to you:     Okay, but I know this is just an excuse, and you are just sending me in pursuit of some black chickpeas!!! so don't think that you can fool me!   Is there any idiom, expression or proverb that conveys the same meaning or connotation in English (i.e. to make or ask someone to do an errand just in order to buy time so that you can deal with some of your private affairs)?   I have found to be/go on  but I'm not sure if I can use it or not. I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity? My office is a converted garage about 30 feet from my house.   I have a coaxial cable already running from my house to the office.   I want to attach a Ethernet cable from my DSL modem, in my office, to a computer in my house.   Since I already have the coaxial cable installed, I bought two coax to Ethernet converters (NVA-P-860-BP78 Video Balun), but when I hooked them up I am getting no signal. It was my understanding that an Ethernet signal could travel over a coaxial cable.   Is that wrong? Any ideas why this isn't working? I want to use an old drive as a secondary hard drive without reformatting. But this old hard drive contains my old Vista installation which takes up gigs of space.  I tried changing ownership and it works to some extent but I can not delete anything in Windows/System32 folder.  Is there an easy way to change permissions so I can delete everything?    The solution is to use Takeown as Wil suggested and use cacls to take full control of the files. Since cacls doesn't seem to be recursive, I fired up cygwin and ran this command to recursively change all the files:  for d in $(find directory-of-interest -type d);do   cacls $d'/*' /e /G username:F; done   where directory-of-interest is in my case /cygdrive/g/Windows and username is my username.  I had to manually run it on directories that had spaces in its name. But after all this, I can now delete the directory. Warning: it seems to me that cacls can potentially royally screw things up if improperly used. Is there really no way to power a DSLR by USB? Is there an easy way to remove overlapping faces? In the example remove doubles doesn't work  You might be able to do the following: A to select all faces. X &gt; Only Faces. F to create faces, but after deleting the faces I'm left with something like this.  If I press A to select all and then press F to create faces then there are still overlapping. Blend file Find a file when you know its checksum? Does any test actually prove causation? How do keyboard input and text output work? How to end paragraph text in an inverted pyramid shape? In many old books, the final paragraphs of chapters end in an inverted pyramid shape, like this:    I think this is beautiful, so I would like to imitate this for my book. I've found the code for a triangle in this post   \documentclass{article} \usepackage{microtype,shapepar}  \def\triangleupshape{% {0}% {0}b{0}\\% {8.66}t{-5}{10}\\% {17.32}t{-10}{20}\\% {17.32}e{0}% }      \begin{document} \shapepar{\triangleupshape}A ipsum dolor sit amet, consectetuer    adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. \par \end{document}   Which results in this:    But I'd like it to be inverted and the first part of the paragraph need to be full line width. Only the last few lines should end in an inverted pyramid like in the example above.  Is there a way to do this? Thanks in advance for your help. "Literally" and "Decimate" misuse Recently I've heard American TV commentators say "[a person] was literally decimated" and "[a Senator] was literally thrown under the bus". In the first case I think the person was not actually 10% killed, but in the second, I believe they meant that 57 members of the US Senate carried #58 onto Constitution Avenue and threw him under a (hopefully moving) bus.  Are usages like these normal or acceptable now? I find them grating, myself. Code breaks when updating jQuery due to changes in toggle