Prove If $S \subset R$ is a nonempty set, bounded from above, then for every $\epsilon &gt;0$ there exists $ x\in S$ such that $(\sup S) - \epsilon &lt; x \le \sup S$  How can I prove for this.. ?   I always use this proposition, but I dont know how to prove.
I'm suppose to prove the following...  Let $A$ be a nonempty subset of $\mathbb{R}$. If $\alpha = \sup A $ is finite show that for each $ \epsilon &gt; 0 $ there is an $a \in A $ such that $\alpha - \epsilon &lt; a\leq \alpha $.  This is what I have so far...  Proof: We claim that $\alpha = \sup A $   Then by definition of a supremum $\alpha $ is the least upper bound of set A  Thus $\forall a \in A$,  $a \leq \alpha$   Suppose $\epsilon &gt; 0$   Then $\alpha - \epsilon &lt; \alpha $.   Thus $\alpha - \epsilon &lt; a\leq \alpha$.   I feel like I'm missing some step between my last and second to last step so what am I forgetting? Also is the rest of my proof right?
Add links to official status pages in the read-only mode status message When the site is in read-only mode, I get a message like this:     This site is currently in read-only mode.   Would it be possible to update that message to provide links to these resources on SE's status:   : Official SE tweets on SE's status. : Official blog posts on SE's status. I have a shapefile with 9000 agriculture fields and what I like to do is to determine for each field (=each polygon) the direction of the longest segment.  For instance, the selected polygon in the screenshot has its longest segment from west to east so I would like to mark give the polygon the attribute W (for west) or either E (for east).   Is there any solution to solve my problem in a fast but generalized way?   I do not need very detailed results. It's just for the interpretation of a question.   I'm seeking an answer using ArcGIS 10.2 for Desktop.  To be more precise on what I'm looking for I try to explain more practical: I'm doing winderosion analysis for a specific area with 9000 fields. As result I have a potential susceptibility for each field. I like to check, if the fields with high susceptibility are correlated to a certain orientation of the fields, e.g. check if most of the fields with high risk are oriented in the main winddirection or not.   The search for the longest segment is not really important for me. It was just an idea I had to determine the direction. So forget about the longest segment if there is a way to determine the direction with another method ;) Are there any practical implementation of a homomorphic hashing or signature scheme? A  is a function $H : A \to B$ between two sets with some algebraic structure $(A, *)$ and $(B, \star)$ such that   $H$ is collision resistant, i.e. it is hard to find $x \neq y$ such that $H(x) = H(y)$ and $H$ is a homomorphism, i.e. $H(x * y) = H(x) \star H(y)$.   Are there any practical realizations of such a homomorphic hash function, or even a homomorphic signature scheme (i.e., where we can "add" valid signatures to get a signature of the "sum" of two messages)?  Even better, are there even any libraries implementing this? Find remainder when $f(x^{12})$ is divided by $f(x)$ What is the best collation to use for MySQL with PHP? I'm wondering if there is a "best" choice for collation in MySQL for a general website where you aren't 100% sure of what will be entered? I understand that all the encodings should be the same, such as MySQL, Apache, the HTML and anything inside PHP.  In the past I have set PHP to output in "UTF-8", but which collation does this match in MySQL? I'm thinking it's one of the UTF-8 ones, but I have used utf8_unicode_ci, utf8_general_ci, and utf8_bin before. A linear operator commuting with all such operators is a scalar multiple of the identity. The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.  We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all linear operators over $V$.  One direction is easy to prove. If $T$ is a scalar multiple of the identity, then there exists a scalar $a$ such that $Tv = av$, $\forall v \in V$. Hence, given an arbitrary vector $w$, $$TS(w) = T(Sw) = a(Sw) = S(aw) = S(Tw) = ST(w)$$ where the third equality is possible because $S$ is a linear operator. Then, it follows that $TS = ST$, as required.  I am, however, at a loss as to how to tackle the other direction. I thought that a proof by contradiction,  ultimately constructing a linear operator $S$ for which $TS \neq ST$, might be the way to go, but haven't made much progress.   Thanks in advance! Running this script on: "Red Had Enterprise Linux Server release 5.11 (Tikanga)"  Mail is: version 8.1 6/6/93 (appears quite old).  Have tried various things and cannot seem to get this to work..  echo "Email message" | mail -a /home/user/checks/notprocessed.txt -a /home/user/checks/dirlist.txt -s "Subject" recipient@company.com /home/user/checks   Resulting message:     mail: invalid option -- a   mutt is not installed and I have limited access to this system as it is owned by the company I work for.  I could add in other components, but has to go through change management, and related processes.  uuencode is not available either.  Here is what I am using today and I am trying to add a second file to:  if grep 'Not Empty' /home/user/checks/notprocessed.txt    then    echo "files were found";     SUBJECT="NIGHTLY DIRECTORY LISTING OF FILES SENT TO VENDOR";    FROM="SENDER&lt;sender@company.com&gt;"    # EMAIL="JOHN DOE&lt;recipient@company.com&gt;"    EMAIL="MARY DOE&lt;recipient@company.com&gt;";    /bin/mail -s "$SUBJECT" "$EMAIL" -- -f "$FROM" &lt; /home/user/checks/notprocessed.txt /home/user/checks/dirlist.txt     exit -20 fi   And..  /home/user/checks/dirlist.txt is not attached or included as part of the e-mail.  I would like /home/user/checks/dirlist.txt to be actually an attachment  This actually does work - but includes the notprocessed.txt as the body of the email..  if grep 'Not Empty' /home/user/checks/notprocessed.txt    then    echo "files were found";     SUBJECT="NIGHTLY DIRECTORY LISTING OF FILES SENT TO VENDOR";    FROM="SENDER&lt;sender@company.com&gt;"    # EMAIL="JOHN DOE&lt;recipient@company.com&gt;"    EMAIL="MARY DOE&lt;recipient@company.com&gt;";    /bin/mail -s "$SUBJECT" "$EMAIL" -- -f "$FROM" &lt; /home/user/checks/notprocessed.txt     exit -20 fi Getting higher power output by using more resistors losing ligatures when switching font series or color in the middle of a word What's wrong with this demonstration?:  $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{2}$$ $$A \implies 1 = (-1)^1 = -1$$  So finaly we get  $$1 = -1$$ Add a link to the timeline of a post For the past year I have been quite active on SO and a bit on meta, but I had never heard of the timeline of a post until today in a comment on meta.  There was an  two years ago of this feature as "experimental". ( for that question).  This feature was , answer: this exists, but the link has to be built manually.  I think this could be quite interesting sometimes and I would like to use this, but I will not do so if I have to build the link each time by myself.  This feature has existed for two years and is still "experimental"? Who is using it? Who knows about it?  My feature request: Implement a link in the posts to their timeline. I am sure this nice feature will be used. What is the difference between UNION and UNION ALL? What is the difference between UNION and UNION ALL? Why is my starting gravity low? In the fall of 2011 I made the leap from extract/partial kits to all-grain kits. I'm enjoying AG, but my SG is continuously below the expected. I have brewed 5 AG kits and 4 have had a low SG.   My assumption is my sugar extraction during mashing (or sparging) isn't optimal. I heat my mash water to 165 and mash the grains for 60 minutes. Mashing levels out at 152. For Lautering I heat the water to 175-180. After I drain the initial mash I add the lautering water and let that soak for 10 minutes. the temperature levels out around 170-172. Then I drain into the kettle and being the boil.  I aim for a gentle boil just enough to break the surface tension. I boil for 60 minutes, chill, rack, take a reading and pitch yeast.  Here are some of the numbers  type            expected     actual  kit st paul porter  1.052        1.039   http://www.northernbrewer.com/shop/st-paul-porter-all-grain-kit.html belgium strong  1.067        1.055   http://www.midwestsupplies.com/superior-strong-ale-all-grain-kit.html troegs hop back 1.063        1.049   http://www.byo.com/stories/recipeindex/article/recipes/90-american-amber-a-pale-ale/2196-troeegs-hopback-amber-ale-clone   The fuorth was the Phat Tyre kit from Northern Brewer. Can't remember the SG though. It was low. They taste great, but the numbers weren't "right". What can I adjust so the numbers are more inline with the recipe?   Update Thank you for all the feedback. There are a number of factors for me to review and adjust. Given my current setup I will follow @mdma advice and recirculate the wort once and extend lautering to 30 minutes.  It will be a week or two before I can attempt this, so it will be awhile before I report back again.  Update #2 I now have some awesome local contacts who are helping me out with technique and trouble shooting. Another (major) issue I could be experiencing is draining the wort way too fast. According to my friend. The drain should be low and slow after cycling a gallon or 2 of wort. sparging should take ~2 hours. for each gallon drained plato/brix should drop 3 points and go no lower than 3-5 points at the end.  the second recommendation is to heat up 3-5 gallons of water to 180 and poor it in prior to/while sparging.  my first attempt using this technique proved successful with the gravity being a lot closer to the anticipated OG. I had some temp issues with the batch, but I was so focused on sparging that I ignored the temp. My guess is I would have been right on if my temp didn't fall so low.  Update #3 I had the chance to brew with my mentor this weekend and he pointed out 2 glaring problems with my mashing technique.   Starting mash temp is too low. I was heating my water to 10-12 degrees above target temp. He recommends 17 degrees. between grain absorption &amp; the tun itself most of the heat dissipates within the first few minutes. This does go against what I originally said. That I was hitting my temp marks. Looking back I was hasty with my readings. I should not dump my sparge water (or lauter?) into the tun while draining the wort. The idea is to compact the liquid into the bottom of the tun thus compacting the sugars. I am stirring up the mash when I dump all sparge water in. We kept our sparge water 1-2 inches above the grain bed and our final results we excellent.   A 3rd recommendation he made was to lower the pH of the water. tap water is ~7 and the ideal mash pH is 5.2. He added a solvent, which I cannot remember the name of, which did this for us. I'll be picking up some of that this weekend.  Thanks to all who helped me out with this problem. What is the easiest way to swap occurrences of two strings in Vim? How to reference sections, equations, and figures in a separate document (e.g., supplemental information)? I have a main document (manuscript) and a supplemental document.   I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized? How to automount a partition on login? I have my musics in another partition of the HD, and I would like that, on login, the partition to be auto-mounted, so I can just open the media player and listen to my music. Is there a way? Does a creature with Lifelink provide lifegain equal to its power, or the defender's toughness? Mysql: Select all data between two dates Why is $H(k\mathbin\Vert x)$ not a secure MAC construction? If $H(m)$ is a secure hash function, can't we implement a MAC using $H(k\mathbin\Vert m)$?  However, it seems the more widely used MACs, such as NMAC and  (both originally defined in ) use a much more complicated scheme. Why is this concatenation scheme insecure? Memory allocation for member functions in C++