This should be a famous question on a (finite) algebra. Let $\mathcal{F}$ be a collection of subsets of $\mathbb{N}$ defined by \begin{align*} \mathcal{F}=\left\{A \subset \mathbb{N} : \lim_{n \to \infty}\frac{\#(A\cap \{1,2,\ldots,n\})}{n} \text{ exists}\right\}. \end{align*} Here, we denote by $\#$ the counting measure. Why $\mathcal{F}$ is a (finite) algebra on $\mathbb{N}$? That is, I would like to know how to prove $A \cup B \in \mathcal{F}$ provided that $A,B \in \mathcal{F}$. This should be equivalent to show that the cesaro mean $\{(1/n)\sum_{k=1}^n\textbf{1}_{A \cap B}(k)\}_{n=1}^\infty$ converges if the cesaro means $\{(1/n)\sum_{k=1}^n\textbf{1}_{A}(k)\}_{n=1}^\infty$ and $\{(1/n)\sum_{k=1}^n\textbf{1}_{B}(k)\}_{n=1}^\infty$. However, I don't feel that this claim holds true.
Let $\Omega=\mathbb{N}$. For each $E\subset\Omega$ let $N_n(E)$ be the cardinality of the set $E\cap \{1,2,\ldots,n\}$. Define $C=\left\{E: \lim_{n\rightarrow \infty} \frac{N_n(E)}{n} \text{ exists}\right\}$. Show that $C$ isn't a field.  I already know that it isn't closed to finite union of non-disjoint set but I can't see why. I saw a post that somebody said &quot;the set of natural numbers whose first digit is 1 doesn't have this limit&quot;. I'm in trouble to see why this limit wouldn't exists in finite unions. Thanks in advance.
Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates I'm having a problem converting $\int\limits_1^2  \int\limits_0^ \sqrt{2x-x^2}   \frac{1}{(x^2+y^2)^2}  dy dx $ to polar coordinates.  I drew the graph using my calculator, which looked like half a circle on the x axis.  I know that $\frac{1}{(x^2+y^2)^2}  dydx$ turns to $ \frac{r}{(r^2)^2}drd\theta$, which would be $ \frac{1}{r^3}$  The region of integration in  $\theta $ is I guess $0 \leq  \theta  \leq  \frac{1}{4}  \pi $, but I'm stuck for r. The lower boundary is $x=1=rcos\theta \rightarrow r=1/cos\theta=sec\theta$, but what is the upper boundary?? Is it  $\sqrt{2rcos\theta-r^2cos^2\theta}$? I tried this but I couldn't integrate my equation...  Could someone please help me out?  Also, how can I draw the graph of $y= \sqrt{2x-x^2} $? I could only know how it looked like by using my calculator... What is the correct pronunciation of the word "solder"? I have been listening to a podcast where the host pronounces the word solder as "sodder" or "sod-der", even "saw-der". Same thing happened when the lecturer of one of my EE classes pronounced the word solder. As for me I pronounce the word solder as "sole-der" with a distinct L and a longish O. Derivation of Newton-Euler equations I am in search of a simplified version of the derivation of  (both translational and rotational) for a rigid body (3D block) that has a body fixed frame and where the center of mass of the body is not at the center of gravity.  I can find elementary derivations for the same system when the center of mass is at the center of gravity, but not for my system in question.    I am using the derivation as background research for a rotordynamics project that I'm working on.  Any help and or references would be greatly appreciated!  :) Why do PBX systems use -48 V? I often see tutorials online that connect various commands with different symbols. For example:  command1 |  command2 command1 &amp;  command2 command1 || command2     command1 &amp;&amp; command2   Others seem to be connecting commands to files:  command1  &gt; file1 command1  &gt;&gt; file1   What are these things? What are they called? What do they do? Are there more of them?    . You can post a link to an android market application by useing a link that looks like:     market://search?q=pname:package    While browsing the market, is it possible to find an app's package name? Or is there a convenient way to get or share a link to a market app? Is there any way to post a link to a market app that normal (non-android) browsers can make use of? Relative log file matching with linux? Shell Script for logging into a ssh server I just started playing Overwatch (played a few games during open beta, but wasn't able to play much). I'm enjoying Mercy, but who (in a group situation) should be getting the damage buff? Additionally, when my team splits up, who should I follow to support? How to remove lines included in one file from another file? Pretty sure that my edit was correct but it was rejected? Add a "flag" button in the revision list Does the derivative of a continuous function goes to zero if the function converges to it? How can I produce an encrypted Swap Partition for 14.04 ? All the advices I saw here are for versions &lt;14.04 and do not work for me.  I tried a lot but nothing lasted longer than a reboot.  On installation I decided to encrypt my home and 9 GiB partition for swap but this swap was never used by Kubuntu. Gparted finds an "unknown filesystem".  I many times formated it as swap, edited fstab and crypttab and used all the tricks mentioned in different answers here. The often mentioned PunyGeek letter does not work for me probably because /etc/initramfs-tools/conf.d/resume does not even exist on my system. reference variable In some text it is given that we can't assign constant values to a reference variable. When I executed such a program I could do it. Is there any condition we can't assign a constant value to a reference variable? Why can a 1 element set be a member of another set but not a subset of it? How to disable "time travel" entirely? Every time I open a question I get this oh so funny 90s feel with scrollers and whatnots. In every question I need to click it off. The button seems to only work for the specific page I load; loading a different page shows the theme again.  Is this an attempt to keep people from using the site for a few days or how do I tell you "please never ever do anything like this if you want me to keep contributing here"? Yes I have no sense of humor if this is an attempt at that.  How do I turn off "time travel" (also known as "back to the future") mode off permanently, on all questions, and preferably network-wide? ParametricPlot3D and Sphere Graphics Words with different meanings in American and British English This is similar to , but not quite the same.  There are quite a few words which have totally different meanings in American and British English and which are likely to cause confusion when heard by non-local speakers.  What are they?  We are looking for the same word with different meanings (for example jumper), and not words which are different between the two dialects (for example truck and lorry). Proving $\frac{x}{y} +\frac{y}{z} + \frac{z}{x} \ge 3$ for positive $x,y,z$ Suppose $x,y,z$ are real positive numbers, prove that: $$\dfrac{x}{y} +\dfrac{y}{z} + \dfrac{z}{x} \ge 3$$ with equality when $x=y=z$.  Can someone help me find an easier solution? I started with assume only two are equal, without loss of generality, $x=y$. Then we get $$1 + \dfrac{x}{z} + \dfrac{z}{x}$$ $$1 + \dfrac{x^2 + z^2}{xz} = 1+ \dfrac{(x-z)^2 + 2xz}{xz} = 1 + 2 + \dfrac{(x-z)^2}{xz} \ge 3$$ To prove the general result I considered using a derivative with respect to $y$ to show that, starting with $x=y$ and increasing $y$ by a tiny amount, will show the equation's derivative as positive. I haven't gotten this to work yet. Any ideas?