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This question is from a set of optional, much harder problems from my first year analysis course, but the subject material is norms on $\mathbb R^K$.  (c) Show that there exists a constant $C &gt; 0$ such that for all $\mathbf x ∈ \mathbb R^k$, $||\mathbf x||\le C||\mathbf x||_\infty$  Hint: Use the standard unit vectors to write $\mathbf x = x_1\mathbf e_1 +. . .+x_n \mathbf e_n$, and use the triangle inequality.  (d) Show that there also exists a constant $c &gt; 0$ such that for all $x ∈ \mathbb R^k$, $||\mathbf x||_\infty \le c||\mathbf x||$  Hint: If it isn’t true, then you can find a sequence $⟨\mathbf x_n⟩$ such that $||\mathbf x_n||$ is bounded but $c_n := ||x_n||_∞ → ∞$. (Why?) Think about the sequence $⟨\mathbf x_n/c_n⟩$. The Bolzano-Weierstrass theorem may be of assistance.  Note that the unspecified norm represents any norm for $\mathbb R^k$. I've done part c), but am stuck on part d). I've had a look for proofs on other questions on MSE, but none of them seem relevant to the hint given, and frequently the terminology is too advanced for me.  I tried considering the case if it isn't true, as suggested, but I can't see how to bound any sequence using that. Any help would be much appreciated. How to show that if $\phi(n)$ equals to n itself, then n must be 1? Not that it's a very good idea to change it, but for fun. According to  post, there are still some problems even after changing entries in /etc/passwd, /etc/shadow, and /etc/sudoers. Any suggestions? How can I improve my color selection skill? I'm a web developer, started to become a web designer. However, I think my understanding of colors is not practical yet. There are tools out there that can give you color harmonies, and more stuff, like , but they don't give you the start point. I mean, how should I know that for this web site, I'd better use green and for that website, choose red. How should I know that?  I think color selection is more like an art, instead of being a science, and I'm like "you should become an color artist, in order to become a web designer".  How can I improve my color selection skill? The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural? let $\gamma$ be a closed continuosly differentiable path in the upper half plane  not passing through $i$. What are the possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$ well the integral can be broken like  $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{(z+i)(z-i)}dz=$$  $$\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-i}dz-\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z+i}dz=$$ by Cauchy Integral Formuale $$f(i)-f(-i)$$,so the second integral is $0$ as it is analytic in the upper half plane, but the first integral iss just $n(\gamma,i)$, which iss the windding number of $\gamma$ around $i$ what more I can say? Thank you. How can I make a perfect sweep with bevel object in Bezier curve(3d) (Pipe) I've been working a bash script to automate the process of opening and closing an LVM over LUKS external hard drive. However I continue to receive errors on line 28:   $(vgchange -a n ; cryptsetup luksClose vol)   The error I receive:  $ sudo ./encryption.sh -l Locking System.............. ./encryption.sh: line 28: 0: command not found System Locked!   Here is the rest of the code for reference:  if [ $1 = -u ] ; then     $(cryptsetup luksOpen /dev/sdb2 vol)     echo "Unlocking.................." | pv -qL 10     $(mount /dev/mapper/vol-lvroot /mnt ; mount /dev/mapper/vol-lvvar /mnt/var ; mount /dev/mapper/vol-lvhome /mnt/home ; mount /dev/sdb1 /mnt/boot)     echo "System Mounted!" | pv -qL 10 #   exit 0 fi  if [ $1 = -l ] ; then     $(umount /mnt/var ; umount /mnt/home ; umount /mnt/boot ; umount /mnt/)     echo "Locking System.............." | pv -qL 10     $(vgchange -a n ; cryptsetup luksClose vol)     echo "System Locked!" | pv -qL 10     exit 0 fi   Other information:   I receive no apparent error when running the command in question by itself interactively. I'm new to writing bash, so any recommendations and constructive criticism is appreciated. The process appears to complete successfully, but I don't understand the error or how to make them go away. Upon further research when running the troubled code alone in interactive mode it does provide output after being executed.   This is given after posting the troubled code interactively.   0 logical volume(s) in volume group "vol" now active Show that if a function is not negative and its integral is $0$ than the function is $0$ Suppose $f$ is continuous, $f(x) \ge 0$ on $[a,b]$ and $\int_a^b f(x) dx=0$. Show that $f(x)=0$ on $[a,b]$.   I know that if $f$ in continuous at $x_0$ and $f(x_0) &gt; 0$, then there exists $\delta &gt; 0$ s.t. $f(x) \ge \frac{1}{2} f(x_0)$ for $|x-x_0| &lt; \delta$ and I think it should help me but I'm stuck at this point. Vectors and Norms If $a$ and $b$ are vectors such that $\|a\| = 4$, $\|{b}\| = 5$, and $\|{a} + {b}\| = 7$, then find $\|2 {a} - 3 {b}\|$.  I couldn't figure out how to start off this problem. I attempted to use $$\cos \theta = \frac{a\cdot b}{\|a\|\cdot \|b\|}$$ But I still don't know what to do. Could someone nudge me in the right direction?  Thank you! Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not.  $P(X|X&lt;Y)$ = ... and  $P(X|X&gt;Y)$ = ...  I am thinking solving this in this way:  \begin{align} P(X|X&lt;Y) &amp;= \frac{P(Y&gt;X|X)P(X)}{P(Y&gt;X)} \\ &amp;= \frac{(1-\Phi(\frac{x-\mu_2}{\sigma_2}))N(\mu_1,\sigma_1)}{\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}\\ P(X|X&gt;Y) &amp;= \frac{P(Y&lt;X|X)P(X)}{P(Y&lt;X)} \\ &amp;= \frac{\Phi(\frac{x-\mu_2}{\sigma_2})N(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})} \end{align}  My questions are:   (1) Whether above solution is correct  (2) How to get the mean and sd for $P(X|X&lt;Y)$ and $P(X|X&gt;Y)$ if they are still normal? Information about (simple) mask wearing (concerning covid19) is a complete mystery to me. On several official places it is not recommended for a person without symptoms to wear a mask. For example:    (time 0:20)  or here:    However no explanation is given - which makes me wonder (and even angry). I think everybody should wear a mask because:   Many persons feel ashamed wearing it, even those with symptoms. If everybody wears it, they do not feel ashamed and wear it. As far as I know, one is (wrt covid19) contagious before developing symptoms. So if everybody wears the mask also these people do.   The two above points act as a collective protection.   Simple mask certainly will lower "getting / spreading illness" probability. Of course the "holes" (in a simple mask) are too big to prevent completely virus to enter (or get out when coughing), but it lowers probability! Even if it lowers the transmission probability by e.g. 5%, it might be a lot. As a hypothetical example: it might reduce R0 from 1.02 to 0.97 and so stop the virus spread so fast.   The videos are full of self-contradiction. In the same video    time 0:50 the lady says I should wear a mask if I am in contact with an infected person... why if it is inefficient?  Question: What are reasons for not recommending wearing a mask? Do I really need to encode '&' as '&amp;'? Fractional Scaling 18.04 LTS Create UEFI bootable USB stick to install Windows AND Ubuntu from it I know how to make a UEFI bootable stick to install Ubuntu (explained ) and I know how to make one to install Windows 7 (, sorry). I guess for Win 8 the procedure would be similar, but I haven't looked into it.  I also know how to make a legacy bootable stick from which I can install Windows and Ubuntu using YUMI. I have failed so far though at creating a UEFI bootable stick that can do this. When I do a UEFI boot from my YUMI multiboot stick it gives me no option to choose between Ubuntu and Windows (hardly surprising I guess) and instead simply UEFI boots Windows (which I find kind of surprising to be honest but maybe this is due to the fact that I added the Windows image after the Ubuntu image and this simply replaced Ubuntu's EFI boot files).  So how would I proceed to create such a bootable stick? Why are PATH variables different when running via sudo and su? On my fedora VM, when running with my user account I have /usr/local/bin in my path:  [justin@justin-fedora12 ~]$ env | grep PATH  PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin:/home/justin/bin   And likewise when running su:  [justin@justin-fedora12 ~]$ su - Password:  [root@justin-fedora12 justin]# env | grep PATH PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin:/home/justin/bin   However, when running via sudo, this directory is not in the path:  [root@justin-fedora12 justin]# exit [justin@justin-fedora12 ~]$ sudo bash [root@justin-fedora12 ~]# env | grep PATH PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/sbin:/bin:/usr/sbin:/usr/bin   Why would the path be different when running via sudo? How do I install PlayOnLinux on Ubuntu? Take an FF body and a crop body, with similar MP rating - and the FF body will give a sharper image of the same object. At least, that seems to be the case - as here:    I assume a low ISO was used in both images.  Why does the FF body, with the same lens and almost the same MP, give an image that is so much sharper?   My best guess is that the diffraction / airy still play a role, and the larger pixels reduce effective overlap - thus increasing contrast/sharpness by displaying less of the neighboring diffraction, as below? Why Was It 'Essential' That Voldemort Kill Harry Potter? ‘So the boy ... the boy must die?’ asked Snape, quite calmly.   ‘And Voldemort himself must do it, Severus. That is essential.’   Deathly Hallows - page 551 - UK Hardcover - chapter 33, The Prince's Tale   Dumbledore tells Snape it is 'essential' that Voldemort be the one to kill Harry, I'm assuming in order for the piece of Voldemort's soul in Harry to be properly killed as well.  But why?   Hermione destroyed the Hufflepuff cup; Ron destroyed the Slytherin locket; Neville killed Nagini; Harry destroyed the diadem and Tom Riddle's diary. So the Horcruxes were not immune to destruction at the hand of someone other than Voldemort. And as it ended up, Harry himself wasn't even a Horcrux, but rather just an unknowing host to a parasitic bit of Voldemort's soul.   Why was it 'essential' that Voldemort be the one to kill Harry in order for the piece of Voldemort's soul to die? Is it possible to programmatically stop and start ArcGIS Server map services using Python?