I have recently used induction in solve a problem. But I am told that the proof is wrong. Below is the problem and solution.  I don't see any issue with this proof. What are the issues with this approach?     The population of every city in US is the same      The above statement is equivalent to: “For every n, if S is any set of   n cities in US, then the population of all cities from S is the same”.      Base Case:     Size of S is 1. There is only one city in S. Thus   population of every city in S is the same.      Inductive Hypothesis:       Let S be a set of m cities and assume that population of every city in S is the same.      Induction Step:       We will prove that if S is a set of m+ 1 cities, then the population   of every city in S is the same.       Let S = {c1, c2, · · · , cm, cm+1}. Consider the following two   subsets of S:        S1 = {c1, · · · , cm},      S2 = {c2, · · · , cm+1}      Note that both S1 and S2 are of size m. Thus by induction   hypothesis:      population(c1) = polulation(c2) = · · · = population(cm),   population(c2) = polulation(c2) = · · · = population(cm+1).      Since c2 appears in both sets, we have      population(c1) = polulation(c2) = · · · = population(cm) =   population(cm+1).      Thus for every set of m + 1 cities, their population is the same. By   induction principle, every city in US has the same population.   Are there any rules to decide whether to use Induction or not? What are the mistakes in the above approach.
Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ?
I'm suppose to prove the following...  Let $A$ be a nonempty subset of $\mathbb{R}$. If $\alpha = \sup A $ is finite show that for each $ \epsilon &gt; 0 $ there is an $a \in A $ such that $\alpha - \epsilon &lt; a\leq \alpha $.  This is what I have so far...  Proof: We claim that $\alpha = \sup A $   Then by definition of a supremum $\alpha $ is the least upper bound of set A  Thus $\forall a \in A$,  $a \leq \alpha$   Suppose $\epsilon &gt; 0$   Then $\alpha - \epsilon &lt; \alpha $.   Thus $\alpha - \epsilon &lt; a\leq \alpha$.   I feel like I'm missing some step between my last and second to last step so what am I forgetting? Also is the rest of my proof right? Computing $\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$ I was asked to evaluate the following sum: $$\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$$  I'm trying to use $$\int_{0}^{\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ However, it doesn't seem to work. Any help is greatly appreciated. Remote Desktop Session Black after Minimize PROBLEM: When I minimize a remote desktop session and restore it, the remote desktop screen shows up black. This only happens when connecting to a particular computer.   DETAILS: If I start clicking around in the black area, portions of the screen will start redrawing and showing up correctly. For example, if I leave a window open in the remote session and click where that window is located on the remote computer, then that window--and only that window--will redraw, and sometimes a portion of that window won't redraw (usually the toolbar).  And to clarify--the window only has to be minimized momentarily, so it doesn't seem to be a timeout issue. Clicking or typing in the remote session still causes the remote computer to respond appropriately. Disconnecting from the session and reconnecting restores the whole screen image, as does clicking all over the place in the black image (causing each section to redraw).  CONFIGURATION: This problem only happens for me when connecting to a particular computer (a W2K Server box configured to allow remote administration) and only with certain client computers. I've tried 7 different client computers with various versions of Remote Desktop (the OSes were: Win2K, Server 2003, Server 2008, Windows 7 RC, 3 XP) and two of them exhibit the problem (one is one of the XP boxes and the other is Windows 7). Those same computers can RDP to other computers without problem.  RESOLUTION ATTEMPTS: I have tried the following:   Disable the LOCAL screen saver as mentioned on  Turned off bitmap caching in the client, as mentioned on many forums. Updated to version 6.1 of the remote desktop client Using mRemote (I doubted this would work since it uses MS's code for connecting to RDP servers) Turning off all video acceleration.   QUESTION: Any ideas on what is causing this? What if the hard disk is partitioned into more than 26 sections in Windows? (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a). Compton scattering vs. photoelectric effect Say a photon hits some atom.  What determines whether there will be a photoelectric effect (photon is absorbed, electron is released) or whether there will be a Compton scattering (the photon is  scattered at some angle, and the electron is released with another direction)? How do I sort a list of dictionaries by a value of the dictionary? I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because . Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$ Show that for $n \geq 5$, the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$.     I'm not sure on how to handle this problem. I know that the commutator subgroup $G'$ will consist of products of commutators $[\sigma_{i}, \tau_{i}]$ for permutations $\sigma_{i}, \tau_{i} \in S_{n}$. However, I don't know where to start. What would be the best way to approach this? From a toggle to an impulsion I'm trying to make a circuit (with no programmable circuits, only logic gates, flip flops and latches) that outputs a short impulsion each time it's input toggles from 0 to 1. Do you have any idea of how to do so? Project Euler Question 14 (Collatz Problem) On my Windows 7 system I installed via  from Fonts.com:   The font does not appear in C:\Windows\Fonts. However, it shows up in certain desktop applications, for example  as: Sabon Next Com Display  My intention was to use the font in a LaTeX document test.tex:  \documentclass{article} \usepackage{polyglossia} \setromanfont{Sabon Next Com Display}  \begin{document} Hello World! \end{document}   Unfortunately, xelatex test.tex fails:  […] Couldn't find `Sabon Next Com Displ.cfg' miktex-maketfm: No creation rule for font "Sabon Next Com Display".   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! fontspec error: "font-not-found" !  ! The font "Sabon Next Com Display" cannot be found. !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................    l.4 \setromanfont{Sabon Next Com Display}                                            % Sabon Next Com Display, SabonNex...  ?    Can XeLaTeX even work with fonts installed via SkyFonts? If not, what could be a workaround? Can a Stack Exchange association bonus be lost? I recently hit 200 rep on Stack Overflow and got a big 100 rep bonus. Most know this as the 'association bonus'.  If I offer a bounty of 50 points and thereby lessen my 'base' score, do I lose that association bonus?  What if I offer a 150 point bounty and then work my way back to 200? I have two JSON objects with the same structure and I want to concat them together using Javascript. Is there an easy way to do this? Why doesn't Stack Apps have its own meta? I have a tag in my html like this:  &lt;h1&gt;My Website Title Here&lt;/h1&gt;   Using css I want to replace the text with my actual logo. I've got the logo there no problem via resizing the tag and putting a background image in via css. However, I can't figure out how to get rid of the text. I've seen it done before basically by pushing the text off the screen. The problem is I can't remember where I saw it. Use variable with TOP in select statement in SQL Server without making it dynamic Clicking on  doesn't connect me with the meta S.Ex.SE site.  EDIT: For some entry points, . Observation courtesy Gilles.  I consider it a pity that the devs have not paid attention to this issue nor have indicated that they noted the problem. While HTTPS is not yet official on this network, I refrain from using unencrypted connections. I can't participate in metas, oh well.  EDIT #2 (2015-11-07) - with the switch to CloudFlare SSL the problem looks finally resolved.  EDIT #3 (2015-11-17) - still getting 403 Forbidden from CloudFlare. D'rats!..  EDIT #4 (2016-01-11) - We're still there. It's 2016 already... Starting a bounty. I'm stuck in a problem of mapping numbers from one range to another. I want to calculate popularity of a web page based on the number of page hits on a scale of 10. The problem is total number of web pages are around 41,00,000 and the web pages for which page visit is given are only 1,70,000. Also, the page visit for a web page is between 20,000 and 1.  Now, I'll try to rephrase the above problem in more mathematical problem, I have 41,00,000 objects for which I need to calculate some factor "po". Out of these 41,00,000 objects, 1,70,000 objects contain some number 20,000 &lt;=y &lt;= 1. Now how can I calculate "po" for each object on scale of 10.  The data mostly looks like,  O1: 20,000 O2: 17,000 O3: 16,000 . . . . . . . O19: 334 O20: 300 . . . . . . . On: 1 and so on......   Mostly, objects have page visits = 1.  I tried calculating page visit for each object/maximum page visit for an object but that gives me very small values(like 1/20,000 is a very small number). I want my resulatant po values to be at some distance from one another so that I can differentiate between them.  Can anyone please suggest me some solution ?  Thanks Cancelling upvote on comment? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click).