For each positive integer $n$ let $f_n : [0,1] \to R$ be a continuous function, differentiable on   $(0,1]$, such that   $$|f_n^{~'} (x)| \leq\frac{1 + |\ln x|}{\sqrt{x}} $$for $0 &lt; x \leq 1$.   and such that   $$−10 \leq \int^1_0 f_n (x)~\mathrm{d}x \leq 10.$$   Prove that $\{f_n \}$ has a uniformly convergent subsequence on $[0,1]$.     I try to use the Arzelà–Ascoli theorem, which requires me to prove $\{f_n\}$ is uniformly bounded and equicontinuous.   From the second inequality, we conclude $\{f_n\}$ is uniformly bounded.   However, I stuck with proving they are equicontinuous.   I try to use the Lagrange theorem so that it suffices to show the $\{f_n^{~'}\}$ is uniformly bounded.   However, the first inequality cannot give us what we want since the function on the right side is not bounded on $(0,1]$. So what should I do?
This is almost surely an Arzela-Ascoli question, since it comes from an old exam of which such problems are quite common. Unfortunately, I can't seem to get it though.  $\{ f_n \}$ is a sequence of functions $[0,1] \to \mathbb{R}$ satisfying $|f_n'(x)| \leq \frac{1 + |\ln (x)|}{\sqrt{x}}$ and $-10 \leq \int_0^1 f_n(x) dx \leq 10$ for every $n$. The question is to show the existence of a uniformly convergent subsequence.  In these kinds of situations, the bounds given usually turn into something nice  to force uniform boundedness. Maybe I could hope to establish equicontinuity using the integral somehow, but I'm not told that the $f_n$ are always positive, so the absolute values would probably screw things up anyway.  Is it an easy problem I'm missing or is this totally the wrong track? If so, what's the right track?
Error message says I can't flag... but yes I can When trying to upvote a comment made on a locked post (ordinary lock, not historical lock which remove the vote arrows and flag altoghether) we are greeted with such error message:        Or in words:     This comment is not eligible for voting or flagging   Being the curious type, I went straight away and tried to flag, fully expecting the same message... just to see the flag dialog open up just fine:        I was able to flag as well.  Can the error message please be changed to just "This comment is not eligible for voting"? Or of course, if it's not the intended behavior really disable the option to flag?  The original post on which I experimented is now deleted, but  you can play with. Can I make any injective function bijective by patching the codomain? A short story about finding an inhabitable planet and meeting aliens So the story goes, humans in a spaceship search for an inhabitable planet, being surprised they found none in the vicinity of Earth. They find one, and it seems uninhabited. At the same time are observed by the aliens, who go down on the planet to meet them. The aliens sit them down and tell them a story about war and conflict that ruled the galaxy, and how all the sentient races in the galaxy got together and got rid of the race that tried to conquer all by blowing up their planets.  In the end you realise man is this terrible conqueror the aliens are talking about. I have a question about making an ending symbol.  It is a LaTex-default that every proof environment has an ending symbol. I want to make every theorem-alike environment such as the example environment have also an ending symbol. But how to manually set this up? Specifically, I want to make all these environments ended by a square symbol.  \newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{pro}[thm]{Proposition} \newtheorem{assump}{Assumption} \newtheorem{axiom}{Axiom} \newtheorem*{conj}{Conjecture} \theoremstyle{definition} \newtheorem{defi}[thm]{Definition} \newtheorem{exam}{Example} \newtheorem{exer}{Exercise} \newtheorem*{rem}{Remark} \newtheorem*{notation}{Notation} \newtheorem*{note}{Note}   I am looking for, with gratitude, a comprehensive setup. Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ How do I change the default wifi network that Ubuntu connects to on startup? There is an unsecured public wifi network in my range, as well as my own home WPA secured wifi that I want to use by default. However, on startup Ubuntu always connects to the unsecured one. Sometimes I forget to reconnect to the other one, which is very dangerous.   I am running a default setup of Ubuntu Destkop 11.10 with Gnome 3. I have looked under network settings but I can't find a 'default' option. Saving an Adobe illustrator file as a pdf - are fonts embedded? If so, why does Adobe illustrator say I don't have the font when I open it? If I have an Adobe Illustrator file, and I save it as a .pdf - it looks like the fonts are included in that .pdf file. (I think this because when I open the .pdf file in Reader it looks the way I expect it to.) However, when I open this same .pdf file on a different computer (or workstation) with Adobe Illustrator - Illustrator says I 'don't have the font I need'. This doesn't make sense to me because - If I can see the document correctly in reader, I would expect it to be there for Illustrator as well. Can someone explain why this is? I am asking because saving a file as a .pdf for print seems to handle most font troubles that come up - but sometimes I need to change the text in a file before I print it... I am hoping there is a way to open a .pdf so that .ai recognizes that the font is in there.  But I am unsure on the details of saving as a .pdf in Illustrator, how fonts are handled, and what is possible. Sunlight Sensitivity looks pretty bad — Disadvantage to attack rolls and Perception checks whenever you, your target, or what you're trying to perceive are in direct sunlight. I hope I'm missing something, since no other PC race has a comparable hindrance.  Are there any ways you have used to mitigate this flaw to make players of this race not at a severe disadvantage (or, if it doesn't need mitigation, explain why it's already balanced without any action required)? How do you secure wood framing to a concrete basement floor? Old TV mini-series with a living brain in a lab Explicit vs implicit SQL joins Is there any efficiency difference in an explicit vs implicit inner join? For example:  SELECT * FROM table a INNER JOIN table b ON a.id = b.id;   vs.  SELECT a.*, b.* FROM table a, table b WHERE a.id = b.id; Extending the User model with custom fields in Django What's the best way to extend the User model (bundled with Django's authentication app) with custom fields? I would also possibly like to use the email as the username (for authentication purposes).  I've already seen a   to do it, but can't decide on which one is the best. How to pass the password to su/sudo/ssh without overriding the TTY? Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$ What is the number of Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$? How to keep any Window "Always on Top"? Some software have an option for their window to be "Always on Top". Is there any way to get the same functionality for any window. Let me give an example where this would be useful.  Suppose I have a small browser window with real time stats from a game. The window is automatically updated at the rate of once every 10 seconds. I would like the small window to be always on top. so that I can just look at it whenever I want to, without actually going to it and at the same time I could continue working in my main application, which is at full screen.  Is there any Windows hack or App which can give me this functionality? Using QGIS 3 graphical modeler I created a model with a variety of input parameters (> 10). When starting the model, a nice user interface is generated. But the arrangement of the GUI elements is totally confused and apparently unpredictable during creating the model. It does neither seem to depend on the arrangement of input parameters in the graphical modeler nor on the chronological order in which the input parameters are added to the model.   Is there a way that let's me arrange the GUI elements in the order I desire? How to use Mathematica functions in Python programs? Can you eradicate a disease without finding a Cure first? If you eradicate a disease in Pandemic without finding a cure first (ie remove all of one colour from the board) does it remain eradicated or can it still come back unless it has also been cured?  The rules are a bit unclear on this point ... Or I missed the rule that explicitly states this. How do I get resolvconf to regenerate resolv.conf after I change /etc/network/interfaces? After updating /etc/network/interfaces with something very similar to below, how do I get /etc/resolv.conf to update?  I tried (as root) resolver -u, service networking restart but they didn't work. I also fixed the symlink for resolv.conf and tried resolver -u again. Finally in frustration I rebooted, which did fix the problem by rebuilding /etc/resolv.conf.  Ubuntu 12.04, 64bit server, all the latest patches installed.  Example /etc/network/interfaces:  iface eth0 inet static     address 192.168.3.3     netmask 255.255.255.0     gateway 192.168.3.1     dns-search example.com     dns-nameservers 192.168.3.45 192.168.8.10 Is there a rule in British English about how to pronounce "either"?