Prove that the sequence $\{f_n\}_{n \in \mathbb{N}}$ is uniformly convergent on a compact space $X$ Let $X$ be a compact space and $f_n: X \rightarrow \mathbb{R}, n \in \mathbb{N}$ are continuous and $f_{n+1}(x) \le f_n(x),\lim\limits_{n\rightarrow \infty} f_n=0, \forall x \in X $.  Prove that the sequence $\left\lbrace f_n\right\rbrace$ is uniformly convergent on $X$.  I'm trying to prove that with $G_n=\left\lbrace x \in X: f_n(x) &lt; \epsilon\right\rbrace \Rightarrow \exists n_0: \forall n \ge n_0: G_n=X$ but I didn't succeed. Any solution is appreciated. Thank you.
Dini's Theorem - Proof. I'd like to prove the following:  If $f_j$ are continuous functions on a compact set $K$, and $f_{1}(x) \leq f_{2}(x) \leq \dots$ for all $x \in K$, and the $f_j$ converge pointwise to a continuous function $f$ on $K$ then in fact the $f_j$ converge uniformly to $f$ on $K$.  Attempt:  Let $g_{j}(x) = f(x) - f_{j}(x)$ for all $j$. Then, since $f_j \rightarrow f$ pointwise, we see $g_j \rightarrow 0$ pointwise.  Now, let $\varepsilon &gt; 0$ . And examine { $x \in K : g_{j}(x) &lt; \varepsilon$ }.  I've been told that the next step should be to  show that { $x \in K : g_{j}(x) &lt; \varepsilon$ } is equal to the intersection of $K$ with some open set $U_j$. But I'm not certain why this is true?   Advice? Insight?
Direction of friction for object that rolls with slipping Let's say you hit a cue ball with a pool stick that causes an impulse. If the pool stick hits the ball above the ball's center of mass, is the direction of friction different than if the pool stick hits the ball below the center of mass? What determines the direction? Assume the ball is rolling with slipping. How to zoom in/out an UIImage object when user pinches screen? Increase close vote weight for gold tag badge holders In order to increase the efficiency with which poor quality questions are closed, it could make sense to have weighted close votes for a small subset of qualified users.   A very good way to measure the qualified users would be to leverage the tag badges. However, it needs to be rationally limited to a set of users that are knowledgeable with the subject. A very good way to measure the top users in a tag would be to look at those who have a gold badge in the tag.  If two users who have a gold tag badge pertinent to a question issue a close vote, the question should probably be closed (or probably will be very soon under the current system). I think it would make sense to allow users with a gold tag badge to have a 3x weight for close votes in those tags.  Please keep in mind, this isn't suggesting full moderator one click closure. It only adds weight to the vote. A single user could not do it on their own. These singular users cast close votes all the time, so that aspect would not change. Some will vote to close, some wont, but the ones who do should have a stronger weight to their vote. Even with 3x it will require 2 of said votes to accomplish, or still 2 other users (meaning 3 agreed it should be closed - which is the same amount of high rep users it takes to delete a question and that has a much greater impact than closure).  tldr;  Allow gold tag badge users to have a weighted close vote in their tag I am doing an animation. The images I have attached show 2 different render layer view node results.    Image one shows a cube on layer 1, and 2 icosphere on layer 2. I have layer 2 set to fast blur. I have to use transparent film and therefore I use the Alpha over nodes in composition.  Image two shows a cube and 1 icosphere. Image 1 should not be showing 2 icospheres, as the left most one is behind the cube.    Image 2 is from the same render result, and is correcting showing only 1 icosphere. However, that icosphere should be blurred as it is on layer 2. It seems for whatever reason I can not have both the correct composition effect and also have every object's visibility render correctly. I can only either have everything work fine, with no composition, or composition with every emission object showing through every other object in the scene.   I am aware that masks would solve this problem, but because of the complicated setup of object materials and particles, masking produces these very strange and suddenly appearing artifacts. Material Indexing also produces odd artifacts and does not consistently effect the appropriate materials.   This is also a new problem, I did not have this issue in the last blender version. So, material indexing is out, masking is out, object indexing is out.       I want to use the compositor to blur certain layers, but every emission object in the scene shows through. It shows through as though it was on x-ray, regardless if it is behind another or inside another object. The first image is an example of this, the left icosphere is behind the cube, and should not be visible. That is the problem. Why is it visible when it shouldn't be? Gradient and Swiftest Ascent I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$)  The standard proof is to note that the directional derivative is $$D_vf=v\cdot \nabla f=|\nabla f|\,\cos\theta$$ which is maximized at $\theta=0$. This is a good verification, but it doesn't really help me understand the result. n tasks assigned to n computers, what is the EX value of a computer getting 5 or more tasks? Remove Gnome Notifications or set Permanent Do not Disturb Remove HTML Tags from an NSString on the iPhone There are a couple of different ways to remove HTML tags from an NSString in Cocoa.   is to render the string into an NSAttributedString and then grab the rendered text.   is to use NSXMLDocument's -objectByApplyingXSLTString method to apply an XSLT transform that does it.  Unfortunately, the iPhone doesn't support NSAttributedString or NSXMLDocument. There are too many edge cases and malformed HTML documents for me to feel comfortable using regex or NSScanner. Does anyone have a solution to this?  One suggestion has been to simply look for opening and closing tag characters, this method won't work except for very trivial cases.   For example these cases (from the Perl Cookbook chapter on the same subject) would break this method:  &lt;IMG SRC = "foo.gif" ALT = "A &gt; B"&gt;  &lt;!-- &lt;A comment&gt; --&gt;  &lt;script&gt;if (a&lt;b &amp;&amp; a&gt;c)&lt;/script&gt;  &lt;![INCLUDE CDATA [ &gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt; ]]&gt; Is "Five dollars is all I have" correct? Is there a standard function in C that would return the length of an array? Is there an issue tracker for already reported Mathematica bugs? If I run a randomForest model, I can then make predictions based on the model.  Is there a way to get a prediction interval of each of the predictions such that I know how "sure" the model is of its answer.  If this is possible is it simply based on the variability of the dependent variable for the whole model or will it have wider and narrower intervals depending on the particular decision tree that was followed for a particular prediction? How does asymmetric encryption work? I've always been interested in encryption but I have never found a good explanation (beginners explanation) of how encryption with public key and decryption with private key works.  How does it encrypt something with one key and decipher it with another key? Sometimes when I write, for example, "the boundaries of segments", I am suggested to use "segments' boundaries" or even "segment boundaries".   Or "the page's content" vs. "the content of the page" vs. "the page content". Then I would like to know is it always preferred to use phrases with apostrophe or adjective?  For example      In our terms, they are mainly textual elements within the page content, which serve as the starting point  of a data region (or a data region's starting point ?)  or a distinctive feature within it How to configure Remote Desktop on my office Ubuntu machine and how to connect to it from my Windows laptop at home? Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$ I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences.   NOTE: subsequential limits are limits of subsequences  Prove: $a_n$ is bounded $\implies \liminf a_n \leq \limsup a_n$   Proof:  Let $a_n$ be a bounded sequence. That is, $\forall_n(a_n \leq A)$.  If $a_n$ converges then $\liminf a_n = \lim a_n = \limsup a_n$ and we are done.  Otherwise $a_n$ has a set of subsequential limits we need to show $\liminf a_n \leq \limsup a_n$:  This is where I am stuck... onKeyUp not firing for ESC in IE A concept of the exterior derivative for functionals test class with Batch and Callout I have a problem with FeedItems in my Test class, it's very weird behaviour Does someone explain to me why this happens?  Callout Class  public with sharing class Callout {  public static void runProccess(Case[] records) {     Database.executeBatch(new CalloutBatch(records), 10); }  public static void send(Case[] records) {     Case[] toUpdate = new Case[]{};       for (case record : records) {         HttpRequest req = new HttpRequest();         req.setEndpoint('callout:test/SOME_URL');         req.setMethod('POST');         req.setHeader('Content-Type', 'application/json');         req.setBody('');         Http http = new Http();         HTTPResponse res = http.send(req);          Map&lt;String, Object&gt; response = (Map&lt;String, Object&gt;) JSON.deserializeUntyped(res.getBody());          toUpdate.add(new Case(Id = (Id) response.get('Id'), Status = (String) response.get('Status')));     }      update toUpdate;  } }   CalloutBatch Class  global with sharing class CalloutBatch implements Database.Batchable&lt;Case&gt;, Database.AllowsCallouts{  private final Case[] records;  global CalloutBatch (Case[] records) {     this.records = records; }  global Iterable&lt;Case&gt; start(Database.BatchableContext BC) {     return records; }  global void execute(Database.BatchableContext BC, Case[] scope) {     Callout.send(scope); }  global void finish(Database.BatchableContext BC) { } }   Test Class  @IsTest  private class CalloutTest { @IsTest static void shouldReaciveData() {     String status = 'Closed';     Case caseRecord = new Case(Type = 'Web', Status = 'New');     insert caseRecord;      //insert new FeedItem(ParentId = caseRecord.Id, Visibility='AllUsers', Type = 'TextPost', Body = '');      System.assertEquals(0, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]);      Test.startTest();     Test.setMock(HttpCalloutMock.class, new CalloutTestMock(caseRecord.Id, status));     Callout.runProccess(new Case[]{caseRecord});     Test.stopTest();      System.assertEquals(1, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]); } }   Mock Class  public with sharing class CalloutTestMock implements HttpCalloutMock {  private Id recordId; private String status;  public CalloutTestMock(Id recordId, String status) {     this.recordId = recordId;     this.status = status; }  public HTTPResponse respond(HTTPRequest req) {     HttpResponse response = new HttpResponse();      response.setStatusCode(200);     response.setBody('{"Id": "' + recordId + '", "Status": "' + status + '"}');      return response; } }   This code works, but when I uncomment insert FeedItem in Test Class I receive an error message:     System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out   Class.Callout.send: line 23, column 1   Class.CalloutBatch.execute: line 20, column 1   FeedItem doesn't have an impact on this functionality, (I don't have any Process Builders, triggers etc. on FeedItem object) launch an exe/process with stdin stdout and stderr?