For a sequence ${f_n}$ of measurable functions with common domain E, show that each of the following functions is measurable: inf{$f_n$},sup{$f_n$},lim inf{$f_n$},lim sup{$f_n$}  At first, I want to know what inf{$f_n$} means.   If {$f_n$} is increasing  function, then I understand {$f_1$}  is the infimum of sequence of function {$f_n$}.   But what if it doesn't? If two candidates have a crossed point, then what is infimum of {$f_n$}?  Secondly proving the statement : Since {$f_n$} is measurable on E, for any real number $c$,   {$x\in E |f_n(x) &gt;c$}(Or $\gt$ can be $\lt,\le,\ge$) is measurable.   Then {$x\in E$| inf{$f_n$} $\ge c$} , since for any $n$,$f_n$ $\ge$ inf{$f_n$}  so, done. Is it right?
I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof:     Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X, \mathcal{M})$, then the functions:      $$g_{1}(x)=\sup_{j} f_{j}(x)$$ $$g_{2}(x)=\inf_{j} f_{j}(x)$$ are   measurable.       Proof:      We have       $$g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}  f_{j}^{-1}((a,\infty])$$ $$g_{2}^{-1}([-\infty,a))=  \bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$$      Where the result follows from the fact that $g_{1}^{-1}((a,\infty])\in  \mathcal{M}$ for all $a \in \mathbb{R} \iff g$ is measurable, and   $g_{2}^{-1}([-\infty,a))\in \mathcal{M}$ for all $a \in \mathbb{R}  \iff g$ is measurable.   I understand the last part of the proof, but how do we know $g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}f_{j}^{-1}((a,\infty])$ and $g_{2}^{-1}([-\infty,a))=\bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$? I would appreciate any help with this.
Catalan numbers and triangulation Assume $C_n$ is the number of triangulations of a polygon with $n+2$ sides. Using a combinatorial proof, show that $(4n+2)C_n=(n+2)C_{n+1}$.  I don't even know where to start with this one. I understand the triangulation after doing some googling, but I don't know where to go from here to prove the above using just a combinatorial formula. Why is it disadvantageous in graduate admissions to have a higher degree? indicates that holders of a master’s degree are held to higher standards when it comes to graduate admissions. The stated requirements for some graduate programs () also outright exclude applicants with doctoral degrees. These seem to be saying that holding a higher degree prior to application is a disadvantage (a fatal one in the case of a PhD). This also applies to private universities: Stanford  (at least in computer science); .  I don't understand why:   If there are two applicants, Alice and Bob, who have achieved largely similar things, it seems logical that both should have a roughly equal chance of being admitted. However, based on JeffE's answer, if Alice has a Masters degree while Bob does not, it's Bob that gets admitted instead of Alice. JeffE's answer even indicates that if Alice has achieved slightly more than Bob, it's still Bob that gets admitted.  If graduate admissions is looking for students that are capable of succeeding in its program, then Alice should be more likely to succeed than Bob since she's already had some experience of graduate study. This should be even more the case if Alice already has a PhD – if she can do it once, she should be able to do it again.    JeffE's answer indicates that since Alice has had more opportunities than Bob, she should also be held to higher standard. However it should also be true that Alice has already achieved what Bob has yet to and in fact might not be able to achieve. Further, JeffE's answer seems to be saying that Alice should do the academic equivalent of financial engineering and pass her graduate-level courses &amp; thesis without getting a master’s degree even if she qualifies for it, which sounds silly.  As for applicants with PhD degrees, my first guess would be that this is because , and the university is acting in the applicant's best interests. However if this is the case, then this is effectively telling the applicant "I know what you want better than you do", which also sounds silly. UC Berkeley's requirements also seem to be saying that if someone with a CV worthy of a tenured position were to apply for a PhD student position, the university will say "no" even though it's clearly better in their best interests (in terms of research output) to say "yes".   Presumably I'm misunderstanding something because if this behavior were irrational, it would already have been changed. What am I missing?  EDIT: I wonder if things are different in programs where the student is involved in research directly without spending time on coursework? How to draw circle tangents and removing the inner lines? I'm new to Adobe Illustrator and I wanted to accomplish this task. How do I remove the inner lines in my drawings? I tried with all the pathfinders effect but it keeps removing my tangent lines (the lines connecting both my circles).    It works like a charm as I followed the selected's answer guide:  Here it is. So, I understand that for a NPN transistor to work the emitter-base junction needs to be forward biased and the collector-base junction needs to be reverse biased.  I understand how current flows from the emitter to the base when forward biased, but I can't seem to wrap my head around how it flows from the emitter, through the base, and into the collector.  From what I understand about the depletion region of a PN junction diode, when forward biased, the electrons in the N-type material are repulsed by the negative charge of the supply (say a battery) and the holes in the P-type material are repulsed by the positive charge from the supply, thus narrowing the depletion region and allowing electrons to flow from the N region to the P region.  I also understand that the holes and electrons are attracted to the source charges when reverse biased, widening the depletion region and blocking any electrons from flowing.  What I don't understand is what happens to the depletion region of the collector-base junction that allows electrons to flow from the emitter to the collector even though the collector-base junction is reverse biased.  Take a simple common emitter circuit for example.  I apparently am not cool enough to post pictures yet so you'll have to use your imagination.  So pretend we have the base connected to the positive terminal on a 1v battery (Vbb).  The collector is connected to the positive terminal on a 10v battery (Vcc).  The emitter and both negative terminals are connected to ground.  Also pretend that there is some kind of current limiting.  How does current get from the emitter through to the collector?  I can see how they are at different potentials, but I don't understand how this overcomes the reverse biasing of the collector-base junction.  Can anyone tell me what actually happens at the junction to allow current flow and how changing Vbb will affect that current? Can NDSolve handle discontinuous data? Why is the determinant of a symplectic matrix 1? Suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 &amp; E_n\\   -E_n&amp;0  \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J.$$ How to figure out $$\det(A)=1~?$$ My approach: I have tried to separate $A$ into four submartix:$$A=\begin{pmatrix}A_1&amp;A_2 \\A_3&amp;A_4 \end{pmatrix}$$ and I must add a assumption that $A_1$ is invertible. by elementary transfromation:$$\begin{pmatrix}A_1&amp;A_2 \\ A_3&amp;A_4\end{pmatrix}\rightarrow \begin{pmatrix}A_1&amp;A_2 \\ 0&amp;A_4-A_3A_1^{-1}A_2\end{pmatrix}$$ we have: $$\det(A)=\det(A_1)\det(A_4-A_3A_1^{-1}A_2).$$ From$$\begin{pmatrix}A_1&amp;A_2 \\ A_3&amp;A_4\end{pmatrix}\begin{pmatrix}0&amp;E_n \\ -E_n&amp;0\end{pmatrix}\begin{pmatrix}A_1&amp;A_2 \\ A_3&amp;A_4\end{pmatrix}^T=\begin{pmatrix}0&amp;E_n \\ -E_n&amp;0\end{pmatrix}.$$ we get two equalities:$$A_1A_2^T=A_2A_1^T$$ and $$A_1A_4^T-A_2A_3^T=E_n.$$ then $$\det(A)=\det(A_1(A_4-A_3A_1^{-1}A_2)^T)=\det(A_1A_4^T-A_1A_2^T(A_1^T)^{-1}A_3^T)=\det(A_1A_4^T-A_2A_1^T(A_1^T)^{-1}A_3^T)=\det(E_n)=1,$$ but I have no idea to deal with this problem when $A_1$ is not invertible. Symfony plugin sfDoctrineActAsTaggablePlugin not working Email searches in Outlook 2013 work as expected. I am trying to refine it by excluding certain folders by default. I was hoping I could uncheck certain folders in Control Panel > Indexing Options > Outlook. Unfortunately, the checkbox is for Outlook itself. I can either search Outlook or not. There is no refinement beyond that. Is there another way to exclude searching of specific folders? I have a requirement to create triggers dynamically after a managed package is installed. I have read that the tooling API can be used to create triggers dynamically however it cannot be used in production. Is it possible to create triggers from a managed package in an org? I want to be able to create the trigger based on some user inputs. Can $n!$ be a perfect square when $n$ is an integer greater than $1$? Can $n!$ be a perfect square when $n$ is an integer greater than $1$?  Clearly, when $n$ is prime, $n!$ is not a perfect square because the exponent of $n$ in $n!$ is $1$.  The same goes when $n-1$ is prime, by considering the exponent of $n-1$. What is the answer for a general value of $n$?  (And is it possible, to prove without Bertrand's postulate. Because Bertrands postulate is quite a strong result.) How to make two people play at the same time? How do I install a different Python version using apt-get? A continuous function that attains neither its minimum nor its maximum at any open interval is monotone Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq  \mathbb R$ , then how to prove that $f$ is monotone ? Can you provide me historical examples of pure mathematics becoming "useful"? I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.  Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that according to the history of mathematics, the math that is useful today was once pure mathematics (I'm not so sure but I guess that when the calculus was invented, it hadn't a practical application).  Also, I guess that the development of pure mathematics is important because it allows us to think about non-intuitive objects before encountering some phenomena that is similar to these mathematical non-intuitive objects, with this in mind can you provide me historical examples of pure mathematics becoming "useful"? Deep Format of a Harddrive on a Linux System I've got an original Asus Eee PC Netbook that I'd like to sell.  When I've sold old computers in the past, I've typically done a deep/secure/zeroing out style format.  How can I do this on a Linux Netbook?  I'm looking for   The command that will let me do this Advice on how I can do this on a device with (seemingly?) non-removable media   Thanks in advance! Signature/date line with fixed width I'm trying to create a signature and date line in a document.  I need the date line to always be in the same spot on the page no matter what name is under the signature line, such as:  ----------------------   ---------------------- Representative           Date  ----------------------   ---------------------- Program Manager          Date   Here is the code I currently have:  \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ User Representative \makebox[2.5in][r]{Date} \\  \vspace{.2in} \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ Program Manager \makebox[2.5in][r]{Date} \   I want to be able to change the names, such as change Program Manager to Joe Foo and have the word 'Date' stay in the exact place.  Any help would be greatly appreciated.  I'm still a newbie at TeX so please forgive my ignorance. Schengen Visa Expiration I have a multiple entry 90 days Schengen visa (category C) from 15 March 2014 till 21 Dec 2014.    With this visa I entered The Netherlands on 4 April 2014 and left there on 13 April 2014.    I am planning to re-enter The Netherlands on 19 Dec 2014 and leave there for London (I have a valid UK visa as well) on 21 Dec 2014, at 7 am from Amsterdam.    Am I permitted to stay in The Netherlands on 21 Dec 2014 or should I leave the country on 20 Dec evening itself? Send authors an inbox message if their question is closed Counting lines, words, characters and top ten words? The + vowel letter I've been told that when "the" is proceeded by a vowel sound, like "apple" or "hour", it's pronounced as "thee" and not as "thu".  But after listening to a couple of songs, I noticed that sometimes this "rule" is not followed. Take for example the two Katy Perry's songs, "Roar" and "The one that got away". In the first she sings "I got thee eye of thu tiger", but in the second she sings "Thu one that got away".  I don't know if it was sang this way to better suit the song melody (I understand nothing about those techniques), but I got confused. What's the correct pronunciation?  Thanks in advance.