Let $\mathbb{Z}(p^{\infty})$ be defined by  $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$  $ with$ $ i \in \mathbb{N} \}$,  I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$.  i try several ways but I have not been successful,   some help ??  thank you
In my recent study of Artinian modules I have been looking at Prüfer $p$-groups (which I will denote by $\mathbb Z_{p^\infty}$ from now on).   I am attempting to prove that $\mathbb Z_{p^\infty}$ is an Artinian $\mathbb{Z}$-module which is not finitely generated.      To do this I am trying to show that every proper non-trivial submodule of $\mathbb Z_{p^\infty}$ is equal to $\left(\frac{1}{p^n}\mathbb{Z_{}}\right)/\mathbb{Z}$ for some $n$.   I am using the following definition of $\mathbb Z_{p^\infty}$: $$\mathbb Z_{p^\infty}=\left\{x\in \mathbb{Q}/\mathbb{Z}\mid \exists n:p^nx=0 \right\}$$ I have read in several places, Atiyah &amp; MacDonald Example 6.3 and Wikipedia to name a few, that the subgroups of $\mathbb Z_{p^\infty}$ are of this form. This would demonstrate that $\mathbb Z_{p^\infty}$ is Artinian. However, I can't seem to find anywhere a proof or example showing why all the subgroups are of this form. I would like to understand this further, as it is not immediate to me that this would be the case, but my attempts to figure out why have so far been futile. I may be thinking too much and missing a simple explanation. I have looked at starting with an arbitrary subgroup $H$ such that $H\neq 0$ and $H\neq\mathbb Z_{p^\infty}$ and seeing if I can deduce anything from that. I got this far, but it doesn't seem to be going anywhere. I have also looked at Sylow's theorems, but I'm not sure if they will be any use to me as $|\mathbb Z_{p^\infty}|=\infty$ (unless I'm wrong about this!).  I would be extremely grateful if anybody could offer a few hints here as I am slowly losing my sanity trying to figure it out!  Thanks,  Andy.
Verification the integral identity $\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$. I have to verify that     $$\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$$   with $|\alpha|&lt;1$. It is my homework and don't know where to begin. How to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? I have installed clean Ubuntu MATE 18.04 LTS and installed Scilab from repository here with    sudo apt-get update sudo apt-get install scilab   After installation I tried to launch it from GUI (Applications → Other → Scilab) - it does not start. Then I tried to launch it from terminal with scilab command. Terminal output is as follows:  $ scilab /usr/bin/scilab-bin: error while loading shared libraries: libjava.so: cannot open shared object file: No such file or directory   so it can't find Java. I reported  about this problem.  I checked existence of libjava.so:  $ dpkg -S libjava.so   openjdk-11-jre-headless:amd64: /usr/lib/jvm/java-11-openjdk-amd64/lib/libjava.so   so it is in place.  What should I do to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? How to copy some, but not all files? So, you can use the * as a wild card for all files when using cp within context of a directory.  Is there a way to copy all files except x file? Why can we use sample variance of the treatment group to estimate population variance of the controls? removeClass and fade animation How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$? I'm about to prove that for any $a,b&gt;0$ and $n\in\mathbb{N},$ the inequality: $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$ holds.   By induction I get: $$\left(\frac{a+b}{2}\right)\cdot\frac{a^n+b^n}{2}$$ $$=\frac{1}{2}\cdot\frac{a^{n+1}+b^{n+1}+ab^n+ba^n}{2}$$ $$=\frac{1}{2}\left(\frac{a^{n+1}+b^{n+1}}{2}+\frac{ab^n+ba^n}{2}\right).$$  Now I have to prove that $$\frac{1}{2}\left(\frac{a^{n+1}+b^{n+1}}{2}+\frac{ab^n+ba^n}{2}\right)$$ $$\leq\frac{a^{n+1}+b^{n+1}}{2}$$ $$\Leftrightarrow a^{n+1}+b^{n+1}\geq ab^n +ba^n.$$ But I don't know how to prove that one, is it possible to be proved by induction ? Thank you. If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$ Let $f:[0,1]\to \mathbb{R}$ is continuous, such that $f(0)=f(1)$. For each $n\in\mathbb{N}$, prove that $\exists x\in [0,1]$ such that $x+\dfrac{1}{n}\in[0,1]$ and $f\left(x+\dfrac{1}{n}\right)=f(x)$.   My approach: The interval $[0,1]$ is connected, then if $f$ is continuous, we have $f([0,1])\subset\mathbb{R}$ is a interval. And I suppose that, I can define a closed path such that $f(0)=f(1)$. Any hint pls, What is the best way (performance wise) to paginate results in SQL Server 2000, 2005, 2008, 2012 if you also want to get the total number of results (before paginating)? Redirect output from all commands to file, not just last To what extent is WhatsApp's statement on secure messaging realistic? I remember being taught that the correct order of adjectives in English was something along the lines of "Opinion-Size-Age-Color-Material-Purpose."  However, it's been a long time and I'm pretty sure I've forgotten a few categories (I think there were eight or nine). Can anyone fill them in? Maximum power transfer and Thevenin I am trying to find the thevenin resistance value and the Vth for maximum power transfer. I have tried to find Rth by calculating R3 and R2 as parallel resistance and same for R4 and R5 and then adding them with R1 but I am not sure if I have done it the right way. For Vth I have tried using KVL but I think I have calculated wrong current values using current divider rule. I calculated the total current by diving V1 and R1. I hope someone can help. I have attached the picture of the circuit. CSS: Special Fluid Layout Problems Suppose that $f : \mathbb{C} \to \mathbb{C}$ is entire. Let $\Omega = \{ z : \frac{1}{2} &lt; |z| &lt; 2\}$. Furthermore, suppose that $ \mathcal{F} = \{f(kz) : k \in \mathbb{C}\}$ is a normal family of analytic functions defined on $\Omega$; the family $\mathcal{F}$ is normal on $\Omega$ iff each sequence from $\mathcal{F}$ has a subsequence converging uniformly on compact subsets of $\Omega$. Prove that $f$ is a polynomial.  I'm studying for my qualifying exam in complex analysis, and this question appears on an old exam. Usually, when I want to show that some entire $f$ is a polynomial, I just try to show that there exists some $N$ so that $f^{(k)}(0) = 0$, all $k \ge N$, but I'm having trouble getting that strategy to work here. I've been looking at the sequence $f(\frac{z}{n})$ trying to make something work, but have not had any success.   I would prefer to receive a hint or two, and then I'll try to post my own solution. Thanks in advance! Proof of "Dual normed vector space is complete" How do I get label of the selected item from a lightning:select drop down list? I would like to get the value of the label of the selected drop down item.  In this case, the label is the Name of the Contact.  &lt;aura:component&gt;  &lt;aura:attribute name="contacts" type="Contact[]"/&gt;  &lt;lightning:select label="Contact" aura:id="contactList" name="contactList"&gt;     &lt;aura:iteration items="{!v.contacts}" var="c" indexVar="index"&gt;        &lt;option value="{!c.Id}"&gt;{!c.Name}&lt;/option&gt;    &lt;/aura:iteration&gt;  &lt;/lightning:select&gt;  &lt;/aura:component&gt;   I know how to get the value:  component.find("contactList").get("v.value");   But how do I get the label? How can I set conditional formatting on a particular cell that depends on another cell's value? I've only seen The Force Awakens once, but I'm as far as I can remember, Phasma doesn't really do much, except:      get taken hostage that one time and disable the shields around the Starkiller after being asked twice   Was she originally supposed to play a bigger role?  I was really expecting to see much more of her, or at least see her in combat. How do I get a list of obsolete packages? I'm trying to figure out a way to get a list of the packages that are no longer available in the repositories that I have enabled. This workstation has been through quite a few versions of Ubuntu and has had many 3rd party repositories added and removed. I'd like to get a list of software that I have from these removed repositories, so I can clean it up or add back the appropriate repositories. I'm using Emacs with AUCTeX to do all my LaTeX. When it comes to syntax highlighting it seems that all mathematics is by default just orange. Do you know of a mode or package that adds some more fine grained syntax highlighting? For example all commands (\foo), matching braces, sub- and superscripts, etc. should get a different color.