Here is a question:  For measurable set $E$ and $F$, with positive measure (in the sense of Lebesgue measure), prove that there is an open interval in the set $E+F$.   I know this is a question from Stein's Real analysis. The point is that I don't know how to prove it using Fubini's theorem. Thanks in advance.
This is a question from Stromberg related to Steinhaus' Theorem:     If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.   I can't quite see how to modify the Steinhaus proof though.
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$ I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$. Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty? This question is related to Exercise 2.15 in Baby Rudin, in which the reader is instructed to "[s]how that Theorem 2.36 and its Corollary become false (in $R^1$, for example) if the word 'compact' is replaced by 'closed' or by 'bounded.'"  The relevant theorem and corollary are as follows:  Theorem 2.36: If $\{K_{α}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{α}\}$ is nonempty, then   $\{K_{α}\}$ is nonempty.  Corollary: If $\{K_{n}\}$ is a sequence of nonempty compact sets such that $K_{n}$ contains $K_{n+1},  (n = 1, 2, 3, ...)$, then   $K_n$ is not empty.  My Question: I realize this question and similar ones have been asked elsewhere on MSE, but I've yet to see a straightforward (to me, at least) answer to my question specifically. Usually, for the "closed" part of the exercise, the following collection of closed sets, $\{K_n \}_{n \in \mathbb{N}}$ is used:  $$K_n = [n, \infty), n \in \mathbb{N}.$$  For, the intersection of every finite subcollection of ${K_n}$ is clearly nonempty. Indeed,  $$ \bigcap_{i=1}^{m} K_{n_{i}} = [N, \infty) $$  where $N = \max\{n_1, n_2, ..., n_m\}$.  Then, the assertion is made that   $$ \bigcap_{n=1}^{\infty} K_n = \emptyset. $$  However, I don't see how the same argument can't simply be used, i.e., that, for example, $N \in  \bigcap_{n=1}^{\infty} K_n$ (or maybe, say, $n+1$). Intuitively, this certainly makes sense, I'd say. For, given the archimedean property, it seems that as $n$ approaches infinity, we can always find an $l$ in each increasingly smaller set such that $n &lt; l &lt; \infty$; and so $l \in  \bigcap_{n=1}^{\infty} K_n$.  What is it exactly that I'm misunderstanding or overlooking here? I will go to France from 17/12/13 – 19/12/13, then I will come back to London. And again I will go back to Portugal and Spain from 26/12/13 to 03/01/14 which I will stay in Spain for the longest period. However, the Spanish consulate told me that I have to request for the visa from the French consulate.  The point is: Do I need to apply for a visa for a single trip two times (from French and Spanish consulate) or a single visa for multiple time (from French consulate). How do I get on a Minecraft PE server when it says 'invalid name'? I am having trouble getting onto a server in Minecraft PE on iOS. When I tap on the person's server it just says invalid name:    I have tried turning off and restarting the iPad, going to home screen and logging back into the server but I have had no luck and no change. Is there a reason why this is so? echo "-e" doesn't print anything I'm using GNU bash, version 3.00.15(1)-release (x86_64-redhat-linux-gnu). And this command:  echo "-e"    doesn't print anything. I guess this is because "-e" is one of a valid options of echo command because echo "-n" and echo "-E" (the other two options) also produce empty strings.  The question is how to escape the sequence "-e" for echo to get the natural output ("-e"). Short story (50s? 60s?) with a man trapped inside a sentient cave Many questions already ask about this topic ( , , etc.) and their answers vary, but they always give exceptions to the apostrophe-s rule, for example:      6.24 The general rule for the possessive of nouns covers most proper nouns, including most names ending in sibilants."   Examples they give include Kansas’s, Ross’s land, and Jones’s reputation. Exceptions include Jesus’  and Moses’.  Which names does this apply to?  Is the Aeneas’  form correct, or is it Aeneas’s instead? UK Train - Can I board at a different station Can I argue that:  \begin{align} \left(\int_a^b f(x) \,dx\right) \left(\int_c^d g(y) \,dy\right) &amp;= \int_a^b f(x) \int_c^d g(y) \,dy \,dx \\ &amp;= \int_a^b \int_c^d f(x) g(y) \,dy \,dx \end{align}  since the same would be true for a product of finite sums? Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why? error: file '/grub/i386-pc/normal.mod' not found. grub rescue&gt;   What can I do? I just sit and stare at it.   I found my old netbook (Dell Inspiron 1010) which I have not used for about four years. I replaced Windows XP with Ubuntu 12.10. I used my bootable USB drive. I installed and rebooted. I got the message that normal.mod is not found.   What should I do? Type exit, reboot, or quit? Should I re-install? Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? Find the sum of two elements in a matrix Given $[a_{ij}]_{10\times10}=\left[ \begin{array}{ccc}   1&amp;2&amp;4&amp;7&amp;...&amp;...&amp;...\\   3&amp;5&amp;8&amp;...&amp;...&amp;...&amp;...\\   6&amp;9&amp;...&amp;...&amp;...&amp;...&amp;...\\   10&amp;...&amp;...&amp;...&amp;...&amp;...&amp;...&amp;\\   ...&amp;...&amp;...&amp;...&amp;...&amp;...&amp;...\\   ...&amp;...&amp;...&amp;...&amp;...&amp;...&amp;98\\   ...&amp;...&amp;...&amp;...&amp;...&amp;99&amp;100\\ \end{array} \right]$, what's $a_{7,3}+a_{7,7}$?  Is there a better way than write down all the elements? It's not easy to find the formula of $a_{i,j}$ Origins of possessive pronouns I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height) I once heard the late John Ciardi (NPR's "A Word in Your Ear") try to explain that the 1920s idiom, "copacetic" (meaning completely satisfactory), entered into the African-American vocabulary in Harlem from the days when Jews and African Americans lived there together.  He argued that copacetic has the same meaning as the Israeli idiom "kol b'seder" which literally means "all is in order."  The problem with that, said my Harlem-raised father-in-law, is that the Jews in Harlem spoke Yiddish and kol b'seder was not used in Yiddish.  The dictionary I've got is not helpful. Can someone come up with a better explanation? The Wifi is working fine but no internet connection in Windows 8 fatal: bad ownership or modes for chroot directory component "/" in SFTP I've been trying to configure sftp in a debian server.sshd_config:  Subsystem sftp internal-sftp  UsePAM no  Match User sftpUser ChrootDirectory /users/sftp/sftpUser ForceCommand internal-sftp AllowTcpForwarding no X11Forwarding no   Directories:   drwxr-xr-x   3 root      root      4096 Oct 20 10:59 users   drwxr-xr-x   3 root      root      4096 Oct 20 11:00 sftp   drwxr-xr-x   2 root      root      4096 Oct 20 11:00 sftpUser   cat /var/log/auth.log  Oct 20 10:58:22 w1 sshd[24634]: Accepted password for sftpUser from 201.156.103.213 port 34106 ssh2 Oct 20 10:58:22 w1 sshd[24636]: fatal: bad ownership or modes for chroot directory component "/"   sftpUser no home, bin/falseand member of user group.  In the client side I'm getting the famous Write failed: Broken pipe and then the connection is dropped.  Commenting the ChrootDirectorycommand in sshd_config makes the connection, but get's the user loose.  What am I doing wrong ? After reading this question about  and asking myself two questions about  and , I realize that these activities are for showing what you found, make the audience be attracted. If they need further explanation, they should read your paper, which has successfully defensed and proved that your work is validated. One important thing when you attempt to attract people is concentrate on the important, which means unnecessary things need to be trimmed. I think the reference is one of those. So, should we put the references in slides/poster when we already have them in paper? Read MIME data using PHP