Assume that $f_n,f:D(0,1)\rightarrow C$ are holomorphic. and $f_n \rightarrow f$ uniformly on every compact set. we need to show that $f'_n \rightarrow f'$ uniformly on every compact set.  I kind of have no idea how to approach this one, its from a test I attended and I'm going for a second go at it and I want to understand the first one, first.
Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove?
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. What is aperture, and how does it affect my photographs? Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ I was generalizing the following integral: $$\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x \hspace{40pt} n\geq 0$$ We can start by noting that $\sin x =\dfrac{e^{ix}-e^{-ix}}{2i}$ and thus $\displaystyle \sin^{2n+1}x=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\sin(2r+1)x$. This implies $$\begin{aligned}\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x &amp;=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\int_{0}^{\infty}\frac{\sin(2r+1)x}{x}\mathrm{d}x \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n(-1)^r\binom{2n+1}{r} \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n\left((-1)^r\binom{2n}{r}-(-1)^{r-1}\binom{2n}{r-1}\right)\end{aligned}$$ Where I have used the well known  $\displaystyle \int_{0}^{\infty} \frac{\sin(2r+1)x}{x}\mathrm{d}x=\int_{0}^{\infty}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$ and the property of binomial coefficients that $\displaystyle \binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$ Since the above sum telescopes, we have $$\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x=\frac{(-1)^n\pi}{2^{2n+1}}(-1)^n\binom{2n}{n}=\frac{\pi}{2^{2n+1}}\binom{2n}{n} ~\forall ~ n\in \mathbb{Z^{+}}$$ I would like to know other methods for evaluating this integral. number of distinct solution of Integral equation Total number of distinct $0\leq x\leq 1$ for which $\displaystyle \int_{0}^{x}\frac{t^2}{1+t^4}dt = 2x-1$   $\bf{My\; Try::}$ Given $\displaystyle \underbrace{\int_{0}^{x}\frac{t^2}{1+t^4}dt}_{\geq 0} = 2x-1\;,$ Where $0\leq x\leq 1$  So we get $\displaystyle x\geq \frac{1}{2}\;\;\;,$ Now  $\displaystyle \frac{d}{dx}\left\{\int_{0}^{x}\frac{t^2}{1+t^4}dt\right\} =\frac{d}{dx}(2x-1)$  So $$\frac{x^2}{1+x^4}=2\Rightarrow \underbrace{x^2+\frac{1}{x^2}}_{\geq 2} = \frac{1}{2}$$  So no real values of $x$  Is my solution is Right, If not then how can i solve it some short way, Thanks How can gravitions exist without violating GR? How do I give a villager trades with custom NBT tags Is it possible, on Debian with Klipper running, to have a separate "clipboard stack" for a particular application?  I'd like the text that I copy in application Y not to be shared with all the other applications running.  All the solutions mentioned at  seems like overkill. How do you tell what brand/make/model of bike I have? Are there other explanations for slow tyre pressure loss? I rode my road bike on Friday and the tyres were at the correct pressure then, around 100psi.  On Tuesday, I jumped on without checking them and quickly realised something was wrong with the rear tyre.  It wasn't completely flat but was fairly soft.  I hadn't gone far so went home and took the car to work instead.  When I got home later, I was surprised to find the tyre, while still soft, hadn't lost all pressure by any means.  It was around 40psi and felt the same as when I left it in the morning.  I decided there must be a slow leak somewhere, but I couldn't find one anywhere, putting the inner under water and going around it methodically looking for bubbles.  I've never failed to find a leak before that way.  I've switched it out anyway because I don't want to mess around with my commute, but I'm curious ... is there another explanation for that amount of pressure loss?  Or must there have been a puncture somewhere that I missed somehow?  The tyres (whole bike) are pretty new and in excellent condition. Can you create a strong blockcipher with small blocksize, given a strong blockcipher of conventional blocksize? Test condition in if command Any idea why echo statement is being executed in both scenarios?  Outputs it worked:  #!/bin/bash  rizwan='' if [ -n $rizwan ]; then   echo "it worked" fi   Replace -n with -z, and it outputs it worked when expecting no output:  #!/bin/bash  rizwan='' if [ -z $rizwan ]; then   echo "it worked" fi Space after LaTeX commands Are amounts of money singular or plural? Are the two sentences correct or is one of them incorrect?     1) Five billion dollars is earmarked for the project.   2) Five billion dollars has been earmarked for the project.   I know dollars are countable. So I think both 'is' and 'has been' are correct here.   Also, do they differ in meaning? Here is the dictionary I have  propertyList = {     "id":           "int",     "name":         "char(40)",      "team":         "int",     "realOwner":    "int",      "x":            "int",     "y":            "int",      "description":  "char(255)",      "port":         "bool",     "secret":       "bool",     "dead":         "bool",     "nomadic":      "bool",      "population":   "int",     "slaves":       "int", }   But when I print it out with "\n".join(myDict) I get this  name nomadic dead port realOwner secret slaves team y x population id description   I know that a dictionary is unordered but it comes out the same every time and I've no idea why. An integer is a $n$'th power if that holds true for all moduli. I have not been able to solve this problem. Any insights would be appreciated!   Let $x, n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_{k}$ such that $x − a_k^n$ is divisible by $k$. Prove that $x = A^{n}$ for some integer $A$. Merging two incoming broadband lines for faster Internet We have a broadband line coming into our building which delivers around 8-10Mbps, which is okay but not great. Fiber is available in our area (30-100Mbps) but it's not available where we are just yet.   Is there some way we could connect two lines (which we would take from two different providers so we would have redundancy if one failed) and merge them into one signal which we could distribute?  Is there some sort of setup that can do this?  Of course we have the option of a leased line but for 12Mbps symmetrical upload/download we'd be looking at around £500 per month compared to £15 per month for normal or fiber broadband.  I made a small pseudo flow chat of the setup below: How many connected components does $\mathrm{GL}_n(\mathbb R)$ have? Why doesn't the command "ls | file" work? I've been studying about the command line and learned that | (pipeline) is meant to redirect the output from a command to the input of another one. So why does the command ls | file doesn't work?  file input is one of more filenames, like file filename1 filename2  ls output is a list of directories and files on a folder, so I thought ls | file was supposed to show the file type of every file on a folder.  When I use it however, the output is:      Usage: file [-bcEhikLlNnprsvz0] [--apple] [--mime-encoding] [--mime-type]         [-e testname] [-F separator] [-f namefile] [-m magicfiles] file ...     file -C [-m magicfiles]     file [--help]   As there was some error with the usage of the file command Basic voltage divider issue in my amateur circuit XBee Once again I need your help in my project. I have a 4,5v powered circuit. I put a voltage divider with 2 resistors in series to reduce my initial voltage of 4.5 V to 3.4 V (to feed an XBee module).  The issue is that if I do not connect the xbee, the voltage divider reduces the input voltage to the desired voltage(3.4 V), but when I connect the XBee module, the resulting voltage divider voltage drops to a value very low (0.85 V), making the XBee malfunction. Surely it is a basic mistake, I am asking something that is very clear, but I'm newbies.   Basic Circuit:  Vin(4.5 V)-----R2(10 kΩ)---------R1(33 kΩ)--------GROUND                        |                        |                  Vout desired (3.4 V)                        |                        |                       XBee After discussion with colleagues regarding the use of the 'var' keyword in C# 3 I wondered what people's opinions were on the appropriate uses of type inference via var?  For example I rather lazily used var in questionable circumstances, e.g.:-  foreach(var item in someList) { // ... } // Type of 'item' not clear. var something = someObject.SomeProperty; // Type of 'something' not clear. var something = someMethod(); // Type of 'something' not clear.   More legitimate uses of var are as follows:-  var l = new List&lt;string&gt;(); // Obvious what l will be. var s = new SomeClass(); // Obvious what s will be.   Interestingly LINQ seems to be a bit of a grey area, e.g.:-  var results = from r in dataContext.SomeTable               select r; // Not *entirely clear* what results will be here.   It's clear what results will be in that it will be a type which implements IEnumerable, however it isn't entirely obvious in the same way a var declaring a new object is.  It's even worse when it comes to LINQ to objects, e.g.:-  var results = from item in someList               where item != 3               select item;   This is no better than the equivilent foreach(var item in someList) { // ... } equivilent.  There is a real concern about type safety here - for example if we were to place the results of that query into an overloaded method that accepted IEnumerable&lt;int&gt; and IEnumerable&lt;double&gt; the caller might inadvertently pass in the wrong type.  var does maintain strong typing but the question is really whether it's dangerous for the type to not be immediately apparent on definition, something which is magnified when overloads mean compiler errors might not be issued when you unintentionally pass the wrong type to a method.