Show that any upper bounded subset of $\mathbb{Z}$ has a unique maximal element
Every non-empty subset of the integers which is bounded above has a largest element.
Does the  tag position in HTML affects performance of the webpage? When should I use "in" or "on"? As it is common with people from my country, I have an immense difficulty with prepositions in English, especially with the use of in and on.   When the preposition indicates the position of the object it is a bit easier to decide which one to use, but as it requires thinking from me, I still say things like "I put the bill in the fridge", "my pencil is on my bag", etc, even though if I think about it, I would know how to say it correctly.  Of course, even in those cases where the preposition indicates position the decision is not always clear, as is the case for example with "on the bus" (you are not on top of the bus, you are inside the bus) or "he is in bed" (you are not inside the bed, you are on top of your bed).  Other cases are even less clear, and I believe there is no real rule to decide what should be said; the person should just know it. Two examples are "in a bad mood", "you are on my mind".  So my first question is whether there is really no rule to help me. The second question refers to two of the cases where I still don't know whether to use in or on:   good luck on/in your exam in/on the list   The last one is especially important to me, as I have asked a number of native speakers, and each time I get a different answer. The only consensus seems to be the case where you say "on top of the list". Apart from that, some people will say that I should say "his name is in the list" whereas others think I should say "his name is on the list". And yeah, a couple of people did tell me that I can use either.  Can someone help me? Since I updated to Ubuntu 17.10 a while ago, my private keys – the which I use for example to access my server via ssh – are not unlocked programmatically after login anymore.    My understanding is, that usually the seahorse client should take care of this, by asking you whether or not to store the key's password when typing them in for the first time.  My first guess was, that seahorse-daemon wasn't running for some reason, but it is:  user@Zeus:~$ ps aux | grep seahorse user    19170  0.0  0.1 432636 26564 ?        Ss   00:07   0:00 seahorse-daemon   My second guess was, that for some reason I have to delete all related passwords stored in seahorse under »Passwords->login«:    I was hoping to force the client to ask me again and then storing them again. But this didn't help either. The client does not pop up to ask me...  Then I found this question, which could be related, but did not help me:    As the management of other passwords work as expected (for example passwords for nautilus, Chromium, Nextcloud etc.) I assume the problem has to do something with ssh-agent...  Can anybody hint me in the right directions, how to solve this problem? Has anything changed in the way how GNOME handles passwords? Maybe some new fancy program failed being installed during the upgrade process?   UPDATE When I add the private key to the authentication agent again with:  ssh-add ~/.ssh/id_rsa   and try to log in, I will be asked to unlock the key only once, after that the key gets unlocked programmatically. But this works only until the next reboot. After a fresh start, I have to add the key again... Is a path the set of points from a to b?  And is the stroke the "definition" (i.e. the color, shape, etc.) of the path?  And, how can you have a path with no stroke?  Is this only because it is a computer drawing program?  In the real world you can't have a path without a stroke can you?  If it matters, I am using Inkscape. How to find the nearest parent of a Git branch? Change all file type associations from one program to another Is it possible to easily change all file type associations that are currently associated with one program to another program in one batch?  In my case, I am trying to change all files types currently associated with Notepad to instead use Notepad++ in Windows 7.  It's a pain to go through Default Programs &gt; Set Associations because after each type the entire list re-populates, taking several seconds.  Is there a way I can do a search and replace in the registry, or something even easier? Session_ID_Remove Issue in API callouts in Salesforce Beryllium Vacuum Sphere Boat/Aircraft Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$ It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?  I got:  $2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\mathrm d\alpha \,\mathrm dr$,  where $x = r \cos(\alpha)$ and $y = r \sin(\alpha)$.  but cannot understand and get the right side. The problem emerged when trying to integrate $\displaystyle \int_0^\infty e^{\frac{-x^2}{n^2}}\,\mathrm dx$ where I tried to change the problem knowing $r^2=x^2+y^2$ but stuck to this part. What is the change in the title called and why is it so? If $\lambda_{max} = n$, then $n\times n$ positive, reciprocal matrix A is consistent At the end of . the author states that  Suppose we have $n \times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $\lambda_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij} a_{jk} = a_{ik}$ where $i,j,k=1,2,...,n$.  I'm not convinced this theorem is true. Could anyone help? Stack Exchange doesn't allow me to answer protected questions because I gave away earned rep in bounties Rendering an animation the way 3D view looks Is there some way to render exactly whats in the camera view in the 3D view? By that I mean just like a screenshot really, all paths, lamps, and cameras, just like things look in the camera view. The reason I can't just take a screenshot is that I need an animation and it would be too cumbersome to take 1000+ screenshots! Ubuntu copy dialogue stops at the last moment for a long time Whenever I try and copy a file to an USB flash disk, the progress bar runs till the end very fast and stops for an indeterminable time before it is complete.  Is there an issue with ubuntu or my flash disk is what I am not able to figure out. All I want is a realistic copy dialogue Is selenium webdriver capable of database testing? After reading so many answers and forums it makes me confused. Some people say: Selenium webdriver is a browser automation tool and it has nothing to do with the database. And some other showing JARs and code that how we can connect selenium to the database and we can test database using selenium webdriver. So I would really like to know if we can do database testing using selenium? If yes then which databases tool it supports like Mysql, SQL server, Oracle... And if selenium is not suitable for database testing then please recommend to me the best tool for database testing. Read Value from URL in PHP How do I get a mesh data-block with modifiers and shape keys applied in Blender 2.8? I'm trying to update an exporter to work with Blender 2.8 but can't seem to get a mesh data-block modified by modifiers or shape keys. It always gives me the base mesh.  I'm currently using Object.to_mesh(). But reading Object.data or bmesh.from_mesh(obj) all returns only the base mesh. So what is the new function to call for getting a mesh data-block with the modifiers and shape keys applied? Evaluate $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n})$ I'm struggling to find the limit at infinity of : $$\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}), n\in\Bbb N$$ I know it is $1$ but I don't understand why this is wrong :  $\sin^2(\pi\sqrt{n^2 + n}) = \sin^2(\pi*n\sqrt{1 + \frac{1}{n}})$.  So $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}) = \lim \limits_{n\rightarrow\infty}\sin^2(\pi*n\sqrt{1 + \frac{1}{n}}) =  \lim \limits_{n\rightarrow\infty} \sin^2(\pi*n) = 0$.  Thanks. Main tag of a question As far as I know, if I have some tags in a question, system will choose the tag which has more questions as main tag (shows in title of page)  But, sometimes this is not correct. For example, I have a question about PhoneGap which applies to both android and iOS. So my question is about PhoneGap, not Android. but system chooses Android as main tag...  Is there any way to manually choose the main tag of a question? if not, Isn't it better to have a feature like this? According to , $[0,1]$ cannot be written as union of countable disjoint closed sets, is the same true about (uncountable) family of disjoint closed intervals ? Bernoulli numbers, taylor series expansion of tan x I found the following formula here: .  Taylor series of $\tan x$:  $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$.  I do not understand what the Bernoulli number means in this formula. My problems are the following ones:  I do not know anything about Bernoulli numbers. I do not know how to expand the Bernoulli number in the formula you see further up. What expansion will be the final result? How did you arrive there? Why did you arrive there? How did you know that you should use the specific Bernoulli-related terms that you used in your answer? What is the purpose of the Bernoulli number in this formula?  A great deal of thanks.