Let $(f_n)$ be a function sequence on $D \subset \mathbb R$ with $f_n \to f^*$ uniform on $D$. Assume $f_n$ bounded for all $n$.   I want to prove the following:  $f^*$ is bounded and  $$  \| f_n \|_\infty \to \| f^* \|_\infty, \quad \sup_{x \in D} f_n(x) \xrightarrow{n\to\infty} \sup_{x \in D} f^*(x)\,. $$    What I have done thus far;  We have that $f_n(x)$ is bounded for each $n \in \mathbb N$. Therefore, for all $n \in \mathbb N$ we can find a $M \in \mathbb R$ (possibly depending on $n$) such that for all $x \in D$ we have $$ |f_n(x)| &lt; M_n\,. $$ As $f_n$ converges uniformly to $f^*$ on $D$, we have $$  \forall_{\epsilon &gt; 0}\exists_{n^* \in \mathbb N}\forall_{n\geq n^*}\forall_{x\in D}: |f_n(x) - f^*(x)| &lt; \epsilon\,. $$  We shall first show that $f^*$ is bounded, therefore we need to prove that there exists an $M^* \in \mathbb R$ such that for all $x \in D$ we have $$  |f^*(x)| &lt; M^*\,. $$  To this end, take $\epsilon = 1$, then for $n \geq n^*$ we have that for all $x \in D$ \begin{align*}  |f_n(x) - f^*(x)| &lt; 1 \iff |f^*(x) - f_n(x)| &amp;&lt; 1\\  \therefore -1 &lt; f^*(x) - f_n(x) &amp;&lt; 1 \\  \therefore -1 + f_n(x) &lt; f^*(x) &amp;&lt; 1 + f_n(x)   \qquad \mbox{but $|f_n(x)| &lt; M_n$, hence} \\  \therefore -1 - M_n &lt; f^*(x) &amp;&lt; 1 + M_n \iff |f^*(x)| &lt; 1 + M_n\,. \end{align*}  Now we have that $f^*$ is bounded by a quantity depending on $n \geq n^*$, which I assume is not enough. At this point I did some research and found  link, which basically says this is sufficient. However, when looking at  link, I would suspect my proof is not enough and we want a value in $\mathbb R$ that bounds $f^*$ independent of any $x$ or $n$.  N.B. I haven't started the second part of the proof yet.
I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete.  The lecturer then proved two things, (i) that $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in \mathbb{R}$.  I don't understand why it's not enough that $f$ is bounded. I think the limit of a sequence of continuous functions is continuous and then if $f$ is bounded, it's in $C_b(X)$. So what is this $\lim f_n \in \mathbb{R}$ about? Many thanks for your help.
Spatial reference for dummies? Can anyone recommend a good reference or tutorial that explains the principles of spatial references, projections, and coordinate systems?   We are a small team of programmers working with GIS technology, and we often get snagged when working with data in different SRS/CS because we don't understand the concepts. I'm trying to use the ant migration tool to retrieve all the Custom Metadata records for a single Custom Metadata Type.  However, I can't figure out the syntax.  I would think that something like this would work:    &lt;Package xmlns="http://soap.sforce.com/2006/04/metadata"&gt;     &lt;types&gt;         &lt;members&gt;State_Sales_Tax.*&lt;/members&gt;         &lt;name&gt;CustomMetadata&lt;/name&gt;     &lt;/types&gt;     &lt;types&gt;         &lt;members&gt;State_Sales_Tax__mdt&lt;/members&gt;         &lt;name&gt;CustomObject&lt;/name&gt;     &lt;/types&gt;     &lt;version&gt;36.0&lt;/version&gt; &lt;/Package&gt;   but no dice. In PHP (>= 5.0), is passing by reference faster? In PHP, function parameters can be passed by reference by prepending an ampersand to the parameter in the function declaration, like so:  function foo(&amp;$bar) {     // ... }   Now, I am aware that this is not designed to improve performance, but to allow functions to change variables that are normally out of their scope.  Instead, PHP seems to use Copy On Write to avoid copying objects (and maybe also arrays) until they are changed. So, for functions that do not change their parameters, the effect should be the same as if you had passed them by reference.  However, I was wondering if the Copy On Write logic maybe is shortcircuited on pass-by-reference and whether that has any performance impact.  ETA: To be sure, I assume that it's not faster, and I am well aware that this is not what references are for. So I think my own guesses are quite good, I'm just looking for an answer from someone who really knows what's definitely happening under the hood. In five years of PHP development, I've always found it hard to get quality information on PHP internals short from reading the source. In the beginning of the series, it's mentioned that Muggle-born wizards/witches are identified due to inadvertent spell-casting. However, it seems like later in the book, disarmed wizards/witches are completely helpless.   What is the explanation for why some spells can be cast without a wand and is there an in-universe explanation for why this isn't done? Also, are there spells that can be cast non-verbally (with/without a wand)? In java SSL over https without certificate My old computer broke down a few days ago, so I'm using a laptop until I can get it fixed.  As a side effect, I no longer have my usual browser cookies, so I had to log back into Stack Exchange.  While doing so, I noticed that I kept having to log in to every Stack Exchange site I visited &mdash; the usual "Welcome back [username], you've been logged in." auto-login banner seemed to be conspicuously missing.  In fact, a quick look at the "network" tab in the developer tools suggested that no requests to stackauth.com were being made at all.  Eventually, I decided to dive into .  Turns out that the code to make the global auth check and bring up the banner is simply:    gauth.checkStackAuth(StackExchange.options.stackAuthUrl);   And, what do you know, running that command from the developer console does auto-log me in.  However, the reason this code isn't being run automatically is because it's disabled in the site config (apparently for all SE sites).  For example, the page I'm typing this question on has the following config options set:  StackExchange.init( {     "locale": "en",     "stackAuthUrl": "https://stackauth.com",     "serverTime": 1408393334,     "networkMetaHostname": "meta.stackexchange.com",     "styleCode": true,     "enableUserHovercards": true,     "site": {         "name": "Meta Stack Exchange",         "description": "Q&amp;A for meta-discussion of the Stack Exchange family of Q&amp;A websites",         "isNoticesTabEnabled": true,         "recaptchaPublicKey": "6LdsB7sSAAAAAAzjgEF_Hd8vXv-C42sa_KyofaGR",         "recaptchaAudioLang": "en",         "enableNewTagCreationWarning": false,         "globalAuthDisabled": true,                // &lt;--- HERE         "isMetaSite": true,         "enableSocialMediaInSharePopup": true,         "useGoogleOAuth2": true     },     // etc.   So, apparently, global auto-login has been deliberately disabled for some reason.  Why is that, and is there any planned timeline for re-enabling it? Can I visit other Schengen countries on a long-term Schengen visa or residency permit? How do I reset all progress in the game "just shapes and beats"? The game  has a very unique intro that I would like to show my friend again for the first time. I'm not sure how to reset all my progress on this game to achieve this though. Any thoughts? The ground was ice-cold, no hint of anyone having lay/laid/lain there   at all.   Which one is the correct option? How do I get the local machine name? I need to find the closure of these sets on the ordered square: $$A = \{(1/n)\times 0 | n\in \mathbb{Z}_+\}$$ $$B = \{(1-1/n)\times \frac{1}{2}| n\in \mathbb{Z}_+\}$$ $$C = \{x\times 0 | 0&lt;x&lt;1\}$$ $$D = \{x\times \frac{1}{2}| 0&lt;x&lt;1\}$$ $$E = \{\frac{1}{2}\times y | 0&lt;y&lt;1\}$$ I know that the closure is just the set itself united with its limit points. For $A$, the set of limit points, I imagine, are $\{&lt;1,0&gt;\}$, that is, the $x$ coordinate is $1$ and the $y$ is $0$. But  is says the reverse. What am I getting wrong? Following the same reasoning, $B' = \{&lt;1,\frac{1}{2}&gt;\}$ $C' = \{&lt;0,0&gt;,&lt;0,1&gt;\}$ What am I doing wrong? Is there a rule forbidding the use of “is” at the end of a sentence? Is there any justification for using “is” at the end of an English sentence, or is there a rule that forbids this? I was reading through the Wikipedia article on  and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the  exactly says      The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.   Now I come to understand that the  singularity is believed to be around 13.8 Billion years ago.  So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?   I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?  Edit: My Bad! Here's how..  A simple google search led me to  which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Should flour be thrown out past its expiry date? Do I need a RHEL subscription to install packages? How to install h.264 decoder? I tried playing videos in movie player but only the audio could be heard. I tried playing the video with dragon player but here also only the audio could be heard.  A message box appeared showing the following message:     Python (v2.7) requires to install plugins to play media files of the following type: H.264 decoder   I tried installing the plugin but it couldn't be installed. It isn't available in the software center too. How can I get this plugin? InDesign: How do I automatically align the bottoms of different columns of text boxes to the same height as the one with the most text? In InDesign, I have some filled text boxes, sized by columns, with the one on the left being three columns wide.   The wide text box is most likely to contain the most text (it's an invoice template).  I will use the 'Fit frame to content' button on the Selection Tool toolbar for automatically vertically resize it to fit the containing text.  How do I make the other three smaller text frames automatically adjust to the same height as the big (and longer) text box? What is the word for act of monotonously demanding for something (many be a toy) repeatedly? What is aperture, and how does it affect my photographs? Selecting all objects that have been specified in Modifiers Could anyone help in making a script for selecting all objects that are specified in the different modifiers that an object has stacked? For example if my object has a Boolean modifier and an Array modifier, which both refer to an empty or another object (in the Array modifier there is for example an Object offset you can specify). Would it be possible to select all those object with one click? Thanks.