$A,B\subseteq(X,d)$ metric space$,\;A\neq \emptyset,\;B\neq\emptyset,\;A\cap B=\emptyset,\;A$ is closed and $B$ is compact.  $d(x,A):=inf\{d(x,a)|\;a\in A\}$  $d(A,B):= inf\{d(b,A)|\;b\in B\}$  Prove that $d(A,B)&gt;0$.  I tried to write definitions and properties (of closed and compact sets) and understood the problem "visually" but I can't write a rigorous proof. Help?
Claim: Let $X$ be a metric space. If $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta&gt;0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$.  Proof. Assume the contrary. Let $\alpha_n\in A,\beta_n\in B$ be chosen such that $|\alpha_n-\beta_n|\rightarrow0$ as $n\rightarrow \infty$.  Since A is compact, there exists a convergent subsequence of $\alpha_n\;(n\in\mathbb{N})$, $\alpha_{n_m}\;(m\in\mathbb{N})$, which converges to $\alpha\in A$.  We have   $$|\alpha-\beta_{n_m}|\leq|\alpha-\alpha_{n_m}|+|\alpha_{n_m}-\beta_{n_m}|\rightarrow0 \;\;\;as\;\;m\rightarrow\infty.$$ Hence $\alpha$ is a limit point of B and since B is closed $\alpha\in B$, contradiction.  Is my proof correct? I feel as though I am missing something simple which would trivialize the proof.
Minecraft LAN "java.net.ConnectException: Connection refused: no further information:" I am trying to play a map on LAN. But every time I try to log on, I get:     java.net.ConnectException: Connection refused: no further information:   This used to work, but for some reason has stopped all of a sudden. I looked it up online, and everything I found said the firewall just wasn't allowing Java through. I followed tutorials to allow Java through the firewall, but even then it won't work. I know for a fact that I'm putting the IP in correctly, and everything seems to be correct, but yet I still get the same error message.  If it helps any, I'm using MC version 1.8.9 (the map requires it), and I don't have any mods. Elements of order 2 in $D_{2n}$ Im new at this abstract algebra stuff and im not comfortable with the proofs techniques yet, so I have a question related to the elements of order $2$ in $D_{2n}$.  Problem:   Prove that $\{x\in D_{2n}|x^{2}=1\}$ is not a subgroup of $D_{2n}$ for $n\geq 3$.   By now, I know that all the elements of the form $sr^{k}$ has order $2$ because I tested by brute force haha, but my real question is how do I prove prove that. How do I have to proceed to prove that $sr^k$ has order 2.  Any help will be really appreciated, thanks so much &lt;3 Beamer and description environment text size What's the proof that the Euler totient function is multiplicative? That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by. I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? I would like to know if there is any man page documenting the construction of the most basic script commands like if/then/else, while, for each, and all the relative switches, like -eq, -e, -ge, and so on. First, my understanding is that the long-term capital gains tax rate is 0% for those whose marginal rate on ordinary income is 10% or 15%, and (ignoring the highest 39.6% bracket) the rate is 15% for everyone else. (See  and .)  My interest is in what happens at the crucial break between the 15% and 25% bracket, which is $73,800 for married/joint returns in 2013 (and presumably will be higher in 2014, though I haven't been able to find out what that is or if it has even been set yet). Let's say a household has $64,000 in wage income, and sells stock for a $10,000 profit.   Does their income for the purposes of establishing their tax bracket go to $74,000? If so, is the entire $10,000 profit subject to the 15% long-term capital gains rate? Is any part of the wage income subject to the higher marginal tax rate?   Since the amount of income subject to the higher marginal rate is, well, marginal, part 3 seems somewhat less important than part 2. The real issue is whether the capital gains rate hits the entire profit, since, in that case, it seems the household should only sell enough stock to generate a $9000 profit, and sell the remainder in a future year. Can I add new methods to the String class in Java? 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). Is it possible to write a regular expression that matches a nested pattern that occurs an unknown number of times? For example, can a regular expression match an opening and closing brace when there are an unknown number of open/close braces nested within the outer braces?  For example:  public MyMethod() {   if (test)   {     // More { }   }    // More { } } // End   Should match:  {   if (test)   {     // More { }   }    // More { } } How much sleep do adults actually need? The common consensus is that most adults should get 8 hours of sleep a night.  However, sleep needs are highly variable, with some people needing more to function properly, and some gaining little benefit past just a few hours. I'm trying to learn more about the statistics here.  I'd imagine sleep needs would fit a normal distribution, with a median of around 8 hours and a standard deviation of around an hour. The  would then put 68% of adults in the 7-9 hour range, 95% in the 6-10 hour range, and 99.7% in the 5-11 hour range, but that's just a guess.  Has this distribution been studied, and where? How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have? I'm running the Storm King's Thunder campaign, and we are at the portion where the PCs get an airship to travel around in, the airship is about 1000ft in the air. There are approximately 5-6 lvl 7 PCs, and they have 3 or 4 hard encounters a day. There are two wizards, a warlock, ranger, paladin, blood hunter, sorcerer, and fighter. For the most part, they have had fun with the different weapons and enemies they have faced; however, I find that one or two spellcasting PCs have been using spells such as mind sliver, polymorph, and hypnotic pattern to defeat more challenging encounters, such as one with a roc and wyvern. They make the creature make saving throws with hypnotic pattern and cause the enemy to fall to its death, and using mind sliver makes it even harder, especially on low wisdom creatures like rocs. Or with the polymorph spell, they cause the enemy to become a fish or chicken and throw it off the edge, and since the damage carries over, it kills the monster. I realize it was legal and made fights funnier, but it does get annoying when it's happening in almost every fight. Are there any suggestions on how to keep them on track without them making every encounter easy? I thought of giving the monsters immunity to these effects, but it just sounds like a bad excuse for me not being prepared. How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Is there a way to consistently increase the vertical spacing between equation lines in a gather (or gather*) environment from amsmath?  Specifically, I want to produce something like  \begin{gather*} a = b \\[2ex] c = d \\[2ex] e = f \end{gather*}   without having to write [2ex] at the end of each equation line.  I can think of some hacks that redefine \\ to automatically add the spacing, but is there some cleaner/better way to achieve this? What is the rule to convert the 'mm' notation to the 'optical zoom' notation? I searched a bit and found this one:  optical zoom = maximum focal length / minimum focal length   For example a 18-55mm lens would have a 3x optical zoom, and a 18-200mm lens would have a 11x optical zoom. Is that right? How do I catalog files on several external hard drives that I want to store off-line? I have quite a few external hard drives that I want to store off-line, and possibly off-site.  The contents is not saved in a structured manner, however; basically  there's lots of project data for lots of projects.  Is there any software for Windows XP that will catalog all the hard drives, so I can search the catalog, find the files, and get the ID of the disk that has the content? Language Options -> Spell Checking in Windows 10 version 1803 How to secure Elliptic Curve ElGamal encryption against known plaintext attacks? If I have an encoding function  $f(x)$ that maps a message $m$ to a point $P$ on a suitable Elliptic Curve $E$ . If I have the public key $Q$ of my recepient then I can encrypt the message as follows:  Choosing a random number $k &lt; n -1 $ where $n$ is the order of the curve E .  Calculating $C = [k] G $ where $G$ is the generator point of the curve E   Calculating $R = [k]Q $ where $Q$ is the recipient's public key   Now the message is encrypted by adding $P$ to $R$ to get          $C_e$ $( = R + P)$ which is sent along with $C$ to the recepient.  DECRYPTION:  Now for the recepient to decrypt my message they compute $R = [privatekey] C $ (where $C = [k]G $) and simply subtract $R$ from $C_e$ to get the encoded point $P$ back which is then decoded by an inverse function $f^{-1}(x)$ to recover $m$.  Now , two problems arise : If the attacker knows the plaintext (suppose if I follow a particular format of data while sending messages ) he could encode his guessed plaintext and subtract it from $C_e$ to recover $ R $ back ! . If I used that same $ R  $ to encrypt further "blocks" of my message then the security of the later parts of my message has been breached!   I realize that doing  something like a scalar multiplication over and over again for every block of data, would be a huge drawback in efficiency and speed.   Is there any way to compute a different $ R $ for every block of data, without using much resources and that too quickly ?   Does reusing the same $R$ for another message break security ?  NOTE: This question is for educational purposes only. It's for the sake of expanding my knowledge on Crypto. AppFabric / WF 4.0 instances stuck despite no bookmark and pending timer