Are there any subsets $A, B$ of $\mathbb R$ such that $\sup(B)>\sup(A)$ but none of the elements of $B$ are upperbounds of $A$? If not, then how would you prove it? I've thought of multiple examples that do not support this but cannot think of how to go about proving it.
If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$ (a) If sup A &lt; sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.  I have argued that if sup A $\lt$ sup B, then choose an $\epsilon&gt;0$ such that sup A +$\epsilon \in B$. Since $a \le $ sup A for all $a \in A$, it follows $a \lt $ sup A + $\epsilon$ hence sup A + $\epsilon $ is an upper bound of A as well as an element of B.  (b) Give an example to show that this is not always the case if we only assume sup A $\le$ sup B.  I am having trouble extending my argument into an example which makes me think that it may not be correct.   Question: Is my argument for (a) correct? If not how would one show this fact? Can you give an example asked for in (b)? Thanks in advance.  I need to understand how I am assuming what I am supposed to prove in my approach. Can someone elaborate on that please? The other question doesn't really address this issue.
Minecraft extra folders? Once I click to run minecraft a small window appears saying "Downloading runtime", then three other folders appear onto my desktop: Game, Native Log, and Runtime. This really annoys me, as I have to keep deleting them after I am done with a session of minecraft. Why is this? It never use to happen before. What made us think that Earth moves around the Sun? Deleting comment removes "Add comment" link on Mobile If I delete a comment on the mobile version of the site (at least if it's the only comment on an answer), the "Add comment" link will vanish along with the comment, so I have to reload to leave another comment.     but got declined as , which is why I'm reposting this (as it may well be a new bug with the same symptoms).  I'm using Firefox 32.0.3 on Android. Suppose $A$ is a 4x4 matrix such that $\det(A)=\frac{1}{64}$ Modifying Score text in Locate Widget of Flexviewer 2.4? I want to add text where it says "score" to make it more clear that this is an address matching score.    I am using the LocateWidget_US.xml and do not think I can do it here.  I think the place to do it may be in the LocateWidget.mxml in this line????   locateResult.content = getDefaultString("resultScoreText") + " " +   (addrCandidate.score % 1 == 0 ? addrCandidate.score : addrCandidate.score.toFixed(1)); Why non-measurable sets exist? What Is the reason behind the fact that $\Bbb R$ is not a complete measure space? Is it only due to the cardinality of  $\Bbb R$? Or other structures on  $\Bbb R$ also play a role? Can one make every subset of  $\Bbb R$ trivially measurable by assigning them to a number of a complete ordered field of the cardinality $\aleph_2$ ? Meaning of convolution? "It really doesn't matter" v "It doesn't really matter" How to add chips from Material Components library to input field in android? How to set a different input method for each app window How can I have a different input method for each app window on windows 8. When I change input language to ENG all app window also uses this method and I don't want that. Like windows 7 that has a different input method for this purpose. thanks "Each of these is" vs. "each of these are" Each of these CDFIs are finding solutions for communities that lack   access to traditional financial products and services, and the NEXT   Awards will accelerate their success.   OR     Each of these CDFIs is finding solutions for communities that lack   access to traditional financial products and services, and the NEXT   Awards will accelerate their success.   The difference is, the first one uses "CDFIs are" and the 2nd sentence uses "CDFIs is"  Which one is correct? Any why? I am looking at  (dual low-voltage power amplifier). On the fourth page, there is a sample circuit; right at the V+ input there is a symbol that looks like a capacitor. However, it looks like it is shaded in. What is that? How do I see that $$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$ for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$? What is the !! (not not) operator in JavaScript? Why is physical space equivalent to $\mathbb{R}^3$? Is there a way I can share the 3.42 GB download of Windows 8.1 from Windows Store? Standard packages for compact SMT crystals There is an old and bog standard form factor for SMT crystals - HC49/US. The problem is that HC49/US takes a lot of real estate on the board.  It's 12mm long.  Another problem with crystals and oscillators in general is that they often become long lead time items.  [This is my experience.]  There are lots of small form factor crystals from various manufacturers.  But, at a glance, it seems that there is no standardization of form factor.  It’s nice when packages are standardized.  It alleviates the long lead time problem.  Crystals from manufacturer B may be used, if crystals from manufacturer A are out of stock.  Are there standard form factors for compact SMT crystals?  Update  Armed with the insights from Gribo and Gustavo, I did a more focused survey.  Found that 12MHz in 3.2x2.5mm package are produced by: Abracon, Citizen, Epson, Fox, ECS, TCS, NDK.  To be included in this list, the part had to be in stock at DigiKey.  It turns out that 3.2x2.5mm package is common across many manufacturers. Given the functional dependencies of a relation is there more then one way to normalize it? How do I free up more space in /boot? I recently obtained a Seagate Expansion 3TB external harddrive. I'm trying to copy over all of my stuff to prepare for reformatting and installing a new OS. A handful of my mp3s did not transfer when I was doing my music folder. I checked them after the fact and the ORIGINAL files became corrupted somehow. I am certain of at least a few that were not corrupted in anyway before doing this. Every now and then during my transfer I got  There was an error copying the file into /media/***/Seagate Expansion Drive. Error splicing file: Input/output error   Each of these files that I check after are corrupted at some point in the duration and will not play. Why did this happen and what can I do to fix it?