I am working through Elementary Analysis by Ross and don't understand part of his proof of Theorem 10.7 (i). This question is in a similar vein as  one, but regarding a different part of the proof.  Theorem 10.7 Let ($s_n$) be a sequence in $\mathbb{R}$. (i) If lim $s_n$  is defined [as a real number, +∞, or −∞], then lim inf $s_n$ = lim $s_n$ = lim sup $s_n$. Proof We use the notation $u_N$ = inf {$s_n$:n&gt;N}, $v_N$ =sup {$s_n$:n&gt;N}, u = lim $u_N$ = lim inf $s_n$ and v = lim $v_N$ = lim sup $s_n$. Suppose lim $s_n$ = +∞. Let M be a positive real number. Then there is a positive integer $N$ so that $n&gt;N$ implies $s_n&gt;M$. Then $u_N$ = inf {$s_n:n&gt;N$} $≥ M$. It follows that $m&gt;N$ implies $u_m ≥ M$. In other words, the sequence ($u_N$) satisfies the condition defining lim $u_N = +∞$, i.e., lim inf $s_n = +∞$. Likewise lim sup $s_n = +∞$. The case of lim $s_n = −∞$ is handled in a similar manner. Now suppose lim $s_n = s$ where $s$ is a real number. Consider $ϵ &gt; 0$. There exists a positive integer N such that |$s_n − s$| $&lt; ϵ$ for $n &gt; N$. Thus $s_n &lt; s + ϵ$ for $ n &gt; N$, so $v_N$ = sup {$s_n : n &gt; N$ } $≤ s + ϵ$ . Also, $m &gt; N$ implies $v_m ≤ s + ϵ$, so lim sup $s_ n$ = lim $v_m ≤ s + ϵ$. Since lim sup $s_n ≤ s + ϵ$ for all $ϵ &gt; 0$, no matter how small, we conclude lim sup $s_n ≤ $lim $s_n$. A similar argument shows lim $s_n ≤ $lim inf$s_n$. Since lim inf $s_n ≤$ lim sup$s_n$, we infer all three numbers are equal: lim inf $s_n$ = lim $s_n$ = lim sup $s_n$.  My question is in regards to this statement towards the end: &quot;Since lim sup $s_n ≤ s + ϵ$ for all $ϵ &gt; 0$, no matter how small, we conclude lim sup $s_n ≤ $lim $s_n$.&quot; Why may we make this assumption? For example, let lim sup $s_n = s + ϵ$. Then since $ϵ &gt; 0$, no matter how small, shouldn't we conclude lim sup $s_n &gt; $ lim $s_n$ in that case? Thank you!
I have been assigned this problem for homework:     Show that, if $a &lt; b + \epsilon$ for every  $\epsilon \gt 0$, then $a\le b$.   I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know how to put it into mathematical terms on paper. any hints?
How would we tell antimatter galaxies apart? Given that  galaxies are theoretically possible, how would they be distinguishable from regular matter galaxies?  That is, antimatter is equal in  and all properties, except for the opposite reverse charge of the particles, identical to regular matter. Hence a star composed of antimatter hydrogen would fuse to anti-helium in an analogous way to our own Sun, and it would emit light and radiation at the same wavelengths as any regular matter star and would cause the same gravitational forces for planetary systems to form as in any other star system.  Hence, what would be a telltale sign if you were observing a galaxy made up entirely of antimatter?  Also, is there any evidence for that half of all galaxies are not made of antimatter -- while general theories currently assume that there is an imbalance of matter over antimatter in the universe, then what is the rationale for not assuming that there is in fact an even balance between the two? What are some scenarios where FTL information transfer would violate causality? I've always heard people saying, "Faster than light information transfer can't happen because it would violate causality!  Effects can precede their causes!"  However, I'm trying to think of a situation where this would happen.  I'm sure it has something to do with relativity or something like that.  I sort of understand that people going faster perceive time slower.  Can someone help illuminate this for me by showing a scenario where causality is clearly violated due to FTL information transfer? Reordering fields (permanently) in file geodatabase using ArcGIS Desktop? Using ArcGIS 10.0, is it possible to reorder the fields in a file geodatabase, and have the order persisted to the database?  We considered exporting to XML, and writing a parser to change the field order, then re-creating a new database from the XML. This seems like a laborious process.  Another idea is to import the existing field definitionss into Excel, re-order them, export to CSV, then write a Python script to create these fields in a new Geodatabase. Does a script like this exist, before we write it from scratch?  Unfortunately the ArcGIS Diagrammer doesn't seem to support field re-ordering.  Are there any scripts or hacks we could use to simplify this process? These are my current disk partitions. /yedek is for some personal backup files. As you can see there is only Ubuntu on my PC. I will install Windows, too. Because of this I want to shrink /dev/sda3 . It currently 546GB and I want to split it: 300GB for / and 246GB for Windows (NTFS) is OK for me.  I'm using GParted for the first time. How can I shrink /?  I don't want to lose any datas from Ubuntu. Multiple solutions for a particle in an infinite potential well I am starting to learn about quantum mechanics and was wondering about the solutions to a question about a particle in an infinite potential well of length $L$.  The solutions are: $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{\pi n x}{L} \right)$$  And I was wondering what is the meaning of $n$. Isn't there supposed to be only one wave function? In that case, $\Psi = \sum_{n=1} ^\infty a_n \psi_n$, but what are the $a_n$? Or maybe it is probabilistic, and we don't know which $\psi_n$ the particle is currently in? (exercise from Tao's analysis book) Proof of a lemma relating to power set of X I'm stuck at one exercise from chapter of sets from Terence Tao's analysis book. I need to proof the lemma:  Lemma: Let $X$ be a set. Then the set $\{Y : Y \:\text{is a subset of}\: X\}$ is a set.  Note: The set $\{Y : Y \:\text{is a subset of}\: X\}$ is known as the power set of $X$, defined as  $2^X $  I can understand why the lemma is true, but I have no clue how to prove it. Furthermore, the author gave a hint that I found very confusing, it follows:  Hint: start with the set $\{0, 1\}^X$ and apply the replacement axiom, replacing each function $f$ with the object $f^{-1}(\{1\})$. If $X$ is an orthogonal matrix, why does $X^TX = I$? It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of $X$ are orthonormal.   Can you explain this better to me? I tried the $2\times 2$ case, but it isn't clear to me why this is true. Algebraic Property inherited by a "lift"? Suppose $K$ is a field, and $K\subset E$ is an algebraic extension, and $K\subset F$ is any extension.  Suppose that $L$ is some field which contains both $F$ and $E$ as subfields.  I want to show that $FE$ (the smallest subfield of $L$ containing both $F$ and $E$) is algebraic over $F$.  Lang mentions that the result is true because an element remains algebraic under lifting, but I can't prove the claim with this.  I interpret his comment to mean that if $x\in E$ is algebraic over $K$, then $x\in FE$ is algebraic over $F$.  This is certainly true.    But this doesn't seem enough if I start with any element of $FE$. OS X El Capitan installation error: AMD-Action:authenticate:SP When doing a clean re-installation of El Capitan (highest supported version) from a bootable USB on an older Mac (MBP, 15-inch, Late 2008), I get this error:  An unexpected error occured while signing in. AMD-Action:authenticate:SP  The error shows up, while logging in with the Apple ID into the App Store in the installer.  I tried with different Apple IDs,  Also tried running these commands in the Recovery Terminal and restarting: csrutil clear csrutil enable Background Video is shown in Rendered mode but not when szene is rendered i have a 3D Object and want to add a background image( if possible even a movie) to this 3D Object. I use Cycles to render and tried a plane with the  image as texture. When i go into render mode, it looks like this(picture with background) and when rendered, it has no background. The key also looks into  a different direction.  Should i even use Cycles for this or use motion tracking? i tried to use that but had several problems.     Rendered mode   When scene is rendered How to pipe the result of 'date' command to 'mv' command I want to rename my file to the current timestamp.I know how to get the current timestamp("date +%s") but how can I pipe it to 'mv' command? How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? Meta Meta Privilege I just took a look at meta today and noticed something weird in the meta privileges. If you look at the meta privilege list, you can see that at . I know that there is no meta for meta, and the link they give you in the privilege page leads you to meta.stackexchange.com, right where you started. Why do they have this privilege? I have a feeling that this is a bad recursive joke. You must be  member of meta to get the privilege of being a member of meta! :) 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 ;) I connect my phone to a Bluetooth speaker to play music. When engage Siri or try to use dictation, the phone wants to use the microphone on the Bluetooth device, which is either too crappy or too far away for it to understand what I am saying.  So, I would like a way to restrict the iPhone to using its own microphone even when connected to a Bluetooth speaker.  I am open to doing this via a setting, buying an app, or jailbreaking or whatever.  Is there a way to do this? Is $\exists x(x=a)$ a valid formula in first order logic with equality? Under , is $\exists x(x=a)$ true in every interpretation in first order logic (FOL) with equality? If so, could $\exists x(x=a)$ be considered a tautology? This question was motivated by  which was closed for lacking clarity. Here I have attempted to add that clarity myself so that the same question can be useful to others who are wondering a similar thing. This is a self answer question. Bibliography error: "Use of \blx@bbl@verbadd@i doesn't match its definition. \verb" Currently having problems with compiling my Master's thesis after downloading the newest version of TeXStudio and MiKTeX. The error message is as follows:  Use of \blx@bbl@verbadd@i doesn't match its definition. \verb   The only other place I found a similar question regarding this error message was found here:   I am not sure how this applies to my problem, but from what I understand this is due to some problem in an external file. If you need the bibliography-entry, I can provide you with that on short notice, though it is rather long (which is why I didn't include it here in the first place).  I am using Mendeley Desktop v1.16.1 to automatically generate my BibTeX database.   I am using biblatex as follows:  \usepackage[backend = bibtex,         style = numeric-comp,         sorting = none,         natbib = true,         doi = false,         isbn = true,         hyperref = true]{biblatex}   I also have some code to change how the bibliography output, but I doubt it is causing the problem, as commenting it out did not help.  Hope anyone knows what causes my problem and how to fix it.  Thanks. Is there a way I can see all the versions that are in the archives that I have configured in sources.list. I can see the last version of each archive withapt-get policy, but how can I see them all?  Is there any way that this can also include PPA and anything in sources.list.d? How to exclude jars generated by maven war plugin? Python List vs. Array - when to use? If you are creating a 1d array, you can implement it as a List, or else use the 'array' module in the standard library.  I have always used Lists for 1d arrays.  What is the reason or circumstance where I would want to use the array module instead?  Is it for performance and memory optimization, or am I missing something obvious?