How do I prove that counting measure is subadditive?  Particularly, I'm confused about, what cases do I need to explore?  $$\mu^* \bigg(\bigcup_{i=1}^{\infty} E_i \bigg)$$  then since each $E_i$ is either some finite natural number or $\infty$, then do I need to induct or something?  Or perhaps one considers:   all $E_i$ are finite at least one $E_i$ is infinite
Can somebody tell me why the counting measure (so, if $S=P(X)$, then $\mu(A)$=infinity if $A$ isn't finite and $\mu(A)=$#$A$ if $A$ is finite) is a measure? (The second property of a measure isn't clear for me).
General identity for a double summation theorem I've been reading a research paper, and I'm interested in generalizing a certain theorem but I can't seem to understand how the following results are derived:  $$\sum_{i=1}^{n}\sum_{j\neq i}^n \frac{1}{z_i-z_j}=0$$  $$\sum_{i=1}^{n}\sum_{j\neq i}^n \frac{z_i}{z_i-z_j}=\frac{1}{2}n(n-1)$$ $$\sum_{i=1}^{n}\sum_{j\neq i}^n \frac{z_i^2}{z_i-z_j}=(n-1)\sum_{i=1}^n z_i$$ $$\sum_{i=1}^{n}\sum_{j\neq i}^n \frac{z_i^3}{z_i-z_j}=(n-1)\sum_{i=1}^n z_i^2 + \sum_{i&lt;j}^n z_i z_j$$ Why was my "shopping list" question closed? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Efficient Way to store images in android Can you trip as part of a Tome of Battle Maneuver? If a ToB Maneuver such as Daunting Strike calls for a melee Attack as part of the maneuver, can this melee attack be resolved as a trip attack/include a trip attempt? Detect whether Excel workbook is already open Adding console.log to every function automatically Continuous weighted random distribution, biased toward one end I am currently contributing to a particle system for our game and developing some emitter shapes.  My uniform random distribution along a line or along a rectangular area works fine - no problem.  But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.  I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient. Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? How to solve 5x5 grid with 16 diagonals? Trying to rig a Map I'm trying to make a "simple" accordion map in Blender 2.79, for animated and still render purposes.  I will map graphics onto each face, and I don't want the faces to get stretched/distorted.  I do however, want to be able to create a hinge-type rig, to be able to fold open and close the map like an accordion without having to manually position each face edge (which would distort the faces anyways).  I looked into using an Armature, but the faces got distorted. I tried to use a Hinge constraint, but I if understand correctly, it would require each flap of the map to be a distinct element - I don't want that. I found solutions involving Weight painting, but I don't think I want any of the map's flaps to be anchored - also there is only one face per element and I don't know if Weight Painting is appropriate.    How can I rig this map properly to open it and close it without any distortion? What kind of "mathematical object" are limits? How to get things off your head in Minecraft Mo' Creatures mod I installed the Mo'Creatures mod into Minecraft, then made a fish bowl. After catching a fish I went to put it on top of a bookshelf in my room, but instead of placing it, the bowl just dropped at my feet. When I tried to pick it up my character put it on his head. Now I'm stuck with a fish bowl on my head and I can't find any mention anywhere of how to get it off. There are a number of other animals from the mod that you can put on your head for some effect, but no mention of how to remove them pretty much anywhere. How do I create an internet shortcut within a folder? Redirect output from all commands to file, not just last I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct? The following is an real analysis qualifying exam problem that I cannot solve:  Suppose $X$ is a Banach space and that $(x_n)$ converges weakly to $x$. Show that $\liminf ||x_n|| \geq ||x||$.  Using the Uniform Boundedness Principle I can show that $\sup_{n \in \mathbb{N}} ||x_n -x||$ is finite. Using Alaoglu's Theorem I can show that some subset of $(x_n)$ converges in norm to $x$. I feel like I am close with this but cannot seem to finish the problem. Does "staff" take a plural verb? Which one of these two statements is correct?     Our staff do ...      Our staff does ...   And is staffs ever correct? How do I add an APN via ADB? For hardware reasons (My menu key doesn't work) I can not add an APN via settings. I read  about using sqlite and  about using code. But I need a more convenient way to do it. I need to do it using adb. I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$ Let $f$ be analytic on an open set $U$, let $a \in U$, and $f'(a)\neq 0$.  Show that  \begin{align*} \frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz \end{align*} where $C$ is a small circle centered at a.  I thought that this is very similar with Cauchy's integral formula. But, I can't apply the Cauchy's integral formula. How to apply that?   Any help is appreciated..   Thank you!