Where $a_n \in C$, intuitively I think this must be true as if are taking an average of almost all arbitrarily small numbers then the average will be arbitrarily small.   let $\epsilon&gt;0, \exists N:|a_n|&lt;\epsilon, \forall n&gt;N$ $$ |(a_1+\cdots+a_n)/n|&lt;\epsilon$$ $$|a_1+\cdots+a_N+\cdots+a_n|&lt;e n$$  I know this is wrong but this was what I was thinking $$|a_1+\cdots+a_N|+(n-N)\epsilon&lt;en$$ Could I have a rough hint on a how to better estimate the value of $ |(a_1+\cdots+a_n)/n|$
I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks!
ERROR running force:source:pull: Entity of type 'CustomApplication' named 'Contact' cannot be found How To Make Clickable Text In Minecraft I am trying to make clickable text in Minecraft in the chat box. It was used in the adventure map, "UP".   The idea is that you are given an option in the text to click, "Yes", or "No". Each word will give an output of a command.  How is this achieved? What sci-fi novel has the moon split into 7 pieces by an asteroid, threatening life on earth? I am looking for the title of a novel that I only caught part of on audio. It takes place within our current time frame and deals with the Moon getting struck and fragmented into (initially) 7 pieces by a high-speed asteroid that they called "The Agent".  Within a few weeks of this episode, scientists figure out the the Moon will continue to break down in a cascade reaction over the next couple of years until the planet gets bombarded by so much debris that it sterilizes Earth. The governments of the world frantically start to build a series of ships using swarming concepts to evacuate the planet as much as possible... and that's all I have heard so far.  Thanks for any info. What is the definition of how to count degrees of freedom? resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here:   Are necessary1 derivatives such as velocities counted as individual DOFs or together with the respective coordinate2? Should complex valued DOFs be counted twice as in "two real-valued DOFs" or once as in "one complex-valued DOF"? (I mean, when one does not want to specify this explicitly)   Please answer with some reference, unless it turns out this is actually a matter of taste rather than a strictly defined thing.    1) I mean those a value of which is required as an initial condition 2) I count fields in QFT as coordinates as well, while space-time-coordinates are parameters to me, if that matters. I know a field actually has $\infty$ (or rather, $2^\aleph$) DOFs itself, but let's say e.g. "one $\mathbb R^3$ continous DOF" in that case I see this a lot, someone in a category of people attempts to shut down anyone with an opinion that is not from the same category. For example, people without children weighing in on parenting; the parent may say something like "well, since you don't have kids and I do, my position is of more validity than yours."  Another example: a drug user (of some kind) telling someone who has never done drugs and does not care to, "until you tried it, you have no right to judge me."  People frequently try to limit the consideration on opinions to those in their same group, while it is perfectly reasonable that the non-drug user, especially if they are a chemist or well read on the subject, may have an equally valid opinion, or even more valid. In fact some might even say that someone NOT in the category has a better chance of being objective and seeing the forest for the trees.  So what is the fallacy of trying to shut down opinions when the source of the opinion is not someone who has personally had the experience under discussion? This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \ge 0$.  Proof:  I will attempt to show that the cauchy criterion is met so $${\sum_{k=n}^m {\sqrt {a_k} \over k}} \le \epsilon $$ for all $n,m \le N_1$. Im disregarding the absolute value since its all positive terms.   Since $\sum{a_n}$ converges, we know that the $\lim \limits_{n \to \infty} a_n =0$ So there exists some $N_2$ such that for all $n \ge N_2$, $$   a_n \lt \ {{(m-n)}^2k^2(\epsilon)^2} $$ I am again disregarding the absolute values because its always positive Let N=max{$N_1,N_2$} then for all $n,m \ge N$ $$\sum_{k=n}^m {\sqrt {a_k} \over k} \le \sum_{k=n}^m {\sqrt {{(m-n)}^2k^2(\epsilon)^2} \over k} \le \sum_{k=n}^m{(m-n)(\epsilon)} = \epsilon $$  Maybe this is just crazy, should I just use Cauchy-Schwarz inequality and forget about this or could I possibly make this method work? Adobe Illustrator: strange behavior while aligning objects I am creating icons with Adobe Illustrator during which process I use the Align tool quite frequently.  All of a sudden, the align tool started behaving quite strangely.  When I select two objects and specify that I want to align to a key object, when I press align it seems to only align halfway to where it should be.  When I press align again it aligns halfway again - after I press it several more times, it finally gets to where it needs to be.  Also, when I try to align to artboard, it pushes the object out to a seemingly random place.  Here is a video showing exactly what I mean:   What setting did I accidentally set to make align behave like this? I am making a skydome and I have a texture hooked up with an emission shader for it, however, since the dome is blue it only emits blue light. Is there a way that I can have the whole sky illuminated and have the light reaching the rest of my scene be a different colour? The interaction of TeX and Lua can be quite delicate at times.  While the former has category codes and tokens, the latter has escape sequences and datatypes.  Mixing the disparate concepts of these two worlds can lead to unexpected behaviour which favours common errors, known as gotchas. In the community wiki answer below you can find an overview of the common mistakes that people make when getting to grips with LuaTeX. As every American teenager "knows" there is no speed limit on the Autobahn. My understanding, however, is that there is still such a thing as excessive speed even on this free-for-all.   So, ignoring the practical "don't be an idiot" and "don't drive faster than you can handle," at what speeds would die Polizei be pulling you over?  Bonus points for translating km/hr into miles/hr. How to find the most famous questions on Stack Overflow? How to find questions with   most comments most votes most viewed   I have seen following question, but seems like there is no easy way.   Is there way to bookmark a page so that the same URL can be used to update data (which will be very seldom) at each hit? Do I need quote marks when I quote myself? Can a US citizen leave and enter the US using a foreign passport? I am a dual citizen. I have both a US passport and a non-US passport. I live in the US. My US passport is currently being processed for renewal. I am unsure whether I will receive my passport in time for a trip that I have scheduled. Could I use my non-US passport to leave the US and return? I have never entered the US with my non-US passport. There is no immigration leaving the US but I still need to scan my passport for ticketing. Will the airline refuse me a ticket? I know that in this situation returning with a non-US passport can be problematic as well, but will I be able to leave at all? Note that I am traveling to a country which neither of my passports are from. Work done in adiabatic process sorry fresh user. I installed about 9 days ago and had issues with Unity disappearing which took 5 days of headaches to fix. It never did fix, I reinstalled.   I set up Plex (media server) and while the server is running and can be accessed by external network devices, I can't add files. Plex says this is because of permissions. I have gone up and down the forums here and at Plex to get an answer. It seems everyone says   chmod -R 755 /media/My\ Book/Videos   will change permissions to allow the folder "Videos" on my external USB 2.0 NTFS 3TB Western Digital drive (which is mounted and accessible through the file manager) as well as every sub-directory and file contained therein to be accessible to user "Plex".  I have used file manager and as soon as you click to change permissions it changes right back. When I chmod with -c it tells me every file is changed (usually 600) to 755 as I commanded it to do. I did   ls -l   to verify and there is no change. (I also used ls -l to make sure I knew who the owner was and it is me (mvl1014). ls -al gives me the same info.)  I swear to god I tried to sudo it and after smashing my head against this keyboard for basically the entirety of my day, I tried it as root. I was scared shi-less but couldn't come up with anything else. sudo su to root and then chmod -R -c /media/My\ Book/Videos  Same result. -c told me every single file was changed and a check in Unity or in terminal confirmed not a darn thing changed.  Now it seems to me that even if that isn't the thing stopping Plex from working there is still a problem with this command not going through.  12.04LTS -- on gen 1 i7 (2.7 GHz I believe) with 6 GB ram   I've attempted to attach a screenshot but I'm told I need 10 reps to do so, which means I'm copy/pasting too.  Please excuse my obfuscation, but Lorem Ipsum seemed unnecessary. And I tried multiple ways to format this but it looks perfect every way and messed up each time in the preview below. (Ugh, I'm a mess.)  $ chmod -R -c 755 /media/My\ Book/Videos  .....  .....  .....  mode of `/media/My Book/Videos/xxxx/xxxx/xxxx.mkv' changed from 0600 (rw-------) to 0755 (rwxr-xr-x)    ls -al shows:  mvl1014@FX6800:/media/My Book/Videos$ ls -al   total 88  drwx------ 1 mvl1014 mvl1014     0 Aug 28 15:17 .   drwx------ 1 mvl1014 mvl1014 20480 Sep  3 14:29 ..  drwx------ 1 mvl1014 mvl1014  8192 Aug 28 15:18 xxxx drwx------ 1 mvl1014 mvl1014 49152 Aug 16 14:02 xxxx drwx------ 1 mvl1014 mvl1014  8192 Aug 27 21:36 xxxx drwx------ 1 mvl1014 mvl1014  4096 Jul 23 19:09 xxxx   I apologize if this has been asked. I looked through this place plus at all the suggested "Similar Questions". The closet thing I got to an answer , but running the provided code with my own UID didn't work either. Mobile web refresher, part 2 After a sizeable amount of work we are releasing a new version of the mobile web site, this time focusing on the question and answer page. We have also fixed or implemented a minimal large part of the issues and requests mentioned in the . Changes Question and answer page  SVG icons for upvote, downvote, favorite, comment upvote white background color for legibility general styling issues with notices (they actually exist now) bounty link is now in question menu (next to flag)   Ask question page  general styling and sizing of element use of placeholders where appropriate instead of page text styling of error messages   Pop-ups  Popups are now inlined in the page instead of being sometimes floating and draggable Many overdue styling issues Close button is SVG Flag indicators are readable Buttons   Misc pages  404 page did not exist. Now it does. Login, logout are formatted Pre-ask and pre-answer warnings are now formatted  Bugfixes  made search magnifying glass bigger fixed language in user timeline so it's consistent with the desktop site added login link on top bar made topbar icons ever so slightly smaller  Restating the obvious: send your &lt;3 to  and   for the most excellent design! Feedback , bugs and suggestions while we move to the next iteration? Please mark bugs with How do I change my username? Some time ago, when I installed Ubuntu, I chose a rather stupid username for my account that I do not want to use anymore. How do I change this (including the name of my home directory, and the name in the terminal) without losing settings for applications? How do I keep permissions and my keys for various authentification (e.g. email, SSH, GPG and more)? What settings could possibly get lost if I changed my username? What are the most appropriate datum and projection I could use for the Gulf coast region - spanning  Louisiana, Mississippi, Alabama, and Florida? I have to analyze block group data along the coast. Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of $R=(\mathcal{U}_n)_n$ such that $\{\mathcal{V}_n|n \in \mathbb{N} \}$ is a discrete subspace of $\beta X$. In particular, there is a sequence $(A_n)_n$ of sets with $A_n \in \mathcal{V}_m$ if and only if $m =n$. If we let $B_n=A_n - (A_0 \cup...\cup A_{n−1})$, then the sequence $(B_n)_n$ is pairwise disjoint and $B_n \in \mathcal{V}_m$  iff $m=n$. If we let $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even. In other words, if $B =\{ \mathcal{V} \in βX|B \in \mathcal{V}\}$, then $B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even. Therefore, we conclude that the sequence $(\mathcal{V}_m)_m$ cannot converge to any point, so the sequence $(\mathcal{U}_n)_n$ cannot converge to any point either.     I would like to know that:      (1) Why "$B_n \in \mathcal{V}_m$  iff $m=n$"? And " $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even "?      (2)$B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even Is infinity a real number?  If not, why not?  I want some very good arguments.   Thanks.  $$\rightarrow\leftarrow\Huge\Huge\Huge\boldsymbol\infty$$