A finite subset of an ordered set contains an $\inf$ and $\sup$ Let S be an ordered set. Let A ⊂ S be a nonempty finite subset. Then A is bounded. Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction.  How do I use induction to prove that the infinum and supremum exist?
Every finite set contains its supremum: proof improvement. Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance.
I have association bonus on superuser.com. After that I've started a bounty worst of 50 rep and earned more than 10 rep. Now I have 89 rep (38 of which is earned at superuser which is more than 10) but still not able to answer protected questions.  So, currently new user with 10 rep can answer protected question and I with 89 (38) rep can't.   How does it work now: after registration on new Stack Exchange site you get 101 rep Association Bonus and are able to answer protected questions only when you have 111+ rep on that site. How should it work: after registration on new stackexchange site you get 101 rep Association Bonus and are able to answer protected questions only when you have earned 10+ rep on that site. What is the difference between the following commands:  su sudo -s sudo -i sudo bash   I know for su I need to know the root password, and for sudo I have to be in the sudoers file, but once executed what is difference?  I know there is a difference between su and sudo -s because my home directory is /root after I execute su, but my home directory is still /home/myname after sudo -s. But I suspect this is just a symptom of an underlying difference that I'm missing. I am attempting to settle a friendly bet.  Would a pendulum swing indefinitely in a hypothetical vacuum (i.e. no air resistance) having a hypothetical frictionless bearing (i.e. no energy lost due to friction) assuming the following   The frictionless vacuum is on Earth (9.8 m/s^2). The pendulum is already in motion and no other external forces other than gravity act on the pendulum. Prove that if $ f\mathrm{(}z\mathrm{)} $ is analytic at the infinity  ,then :  $ {\lim}_{{z}\mathrm{\rightarrow}\mathrm{\infty}}{f}\prime{\mathrm{(}}{z}{\mathrm{)}}\mathrm{{=}}{0} $ How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? What happened to Finn's blaster at Maz Kanata's? When entering, we see that Finn has the blaster that was given to him earlier:    However, when the fighting begins, Finn tells Maz that he needs a weapon and no longer has his blaster with him.  It seems unlikely to have been lost in the first shots, given how it is secured across his body.  Also, Han and Chewie didn't lose their weapons and they are arguably less secure.  Now, when the team gets to the Resistance base, Finn again has his blaster:    So where was this blaster during the battle? GRUB2 does not show Windows 8 option to boot into after Dual Boot Yesterday, I tried to dualboot Ubuntu 16.04 onto my hard drive, creating a new partition for Ubuntu alongside my existing Windows 8. First, I disabled the fast-boot option in Windows 8, and still GRUB did not load, and it went directly to Windows boot-up. Then I did some research and found this article:     Basically, I issued the BCDEdit command in Windows, and entered in the following code:  bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi   When I reboot, I get a basic GRUB2 terminal, and after I type in the exit command, then a second Shell prompt appears, and after typing the exit command again, the GRUB2 menu to select the OS to boot finally appears... with only Ubuntu and Ubuntu memory test as options, not Windows 8.  After booting into Ubuntu, I tried to run the Boot-repair Utility, but that shows this following error:   How do I approach this problem? The Pastebin log is here:   Thanks for the help! How do I set a monitor resolution that is not available in the Display Settings dialog? How does accepting an answer work? Which livestock is the most efficient food source? Displaying attachments on visual force page how to set static ip from bash scriptlinux I want to set static IP using bash script what I would to achieve I want to set the same IP to static which is come from DHCP, for example, my eth0 set to DHCP and it picks IP 51.89.84.82 for eth1 it pick 10.0.0.1 I want to set the same IP to static how can I achieve this in centos7. 1 ) find interface name and its ip , subnet mask , gateway. 2 ) loop each interface and set the same ip , subnet and gateway. so I've recently installed my first Ubuntu, to be precise, a daily build of 16.04, on my new laptop since the hardware was to new for 15.10. Now I'm in the process of setting it up and I want to install php5 and some extensions.  When I execute sudo apt-get install libapache2-mod-php5 I get the error message that there is no installation package available for php5 and libapache2.  The error message is in german, so I doubt that this would help, but anyways here it is. :-) E: For package »libapache2-mod-php5« existiert kein Installationskandidat.  Do I need to add a apt-get repository? Or what am I doing wrong? So I was reading this:  and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any.  So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative?   EDIT Ok, I changed my question a little bit:  Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.)  Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&amp;Lifshitz's mechanics book only consider this kind of change in the Lagrangian? prominently sports purple hair.    Is this her natural hair color?  Natural extensions in the answer depending on the correct answer: Is she wholly human and humans in the Galaxy Far Far Away can have this hair color?   Or is she partly human or not human?   Or does she dye her hair?   Or is there another hair color changing technology? Reviewing a suggested edit from a review queue jumps to the suggested edit queue I was in the first posts review queue and chanced on a post with a suggested edit. (Note: I have enough reputation to review suggested edits.) I clicked on “edit (1)” and was taken into the suggested edits queue. When I approved the suggested edit, I was taken to the next suggested edit, and so on until there were no more suggestions (“There are no items for you to review.”, still in the suggested edits queue). In this situation, I think it would make more sense if I was brought back to that post I was working on in the first posts queue, as soon as I acted on its suggested edit. Alternatively, the suggested edit should be shown in the same kind of pop-up as on a question thread page. The same goes for other queues (late answers, low quality, etc.). Prove norm doesn't come from inner product. Please help me prove this. I'm not sure how to apply the parallelogram law to the norm. On SO, when I'm scrolling through "questions" some have a yellow background.  Is this intuitive to everyone but me?  What is the yellow background trying to tell me about this particular question? According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.  Refer: Einstein's thought experiment of a Physicist in an Accelerated Box.  I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank. prevent php classes from being extends