...without using the fact that a Cauchy sequence of real numbers converges:  If a subsequence of a Cauchy sequence converges, then the Cauchy sequence converges.
Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon &gt; 0$. Take $N &gt;0$ such that for all $n,m &gt; N$, we have   $$d(x_n,x_m) &lt; \frac{\epsilon}{2}.$$  By hypothesis, we can take also $K &gt;0$ such that for all $n_k &gt; K$, we have  $$ d(x_{n_k},x) &lt; \frac{\epsilon}{2}.$$  Put $M = \max \{N,K\}$. Therefore, for all $n,m,n_k &gt; M$, we have  $$ d(x_n,x) \leq d(x_n, x_{n_k}) + d( x_{n_k},x) &lt; \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$  Hence, $x_n \to x$ as desired.  Is this a correct approach? Thank you very much in advance.
What is the real answer to the Birthday question? "How large must a class be to make the probability of finding two people with the same birthday at least 50%?"  I have 360 friends on facebook, and, as expected, the distribution of their birthdays is not uniform at all.  I have one day with that has 9 friends with the same birthday. (9 months after big holidays and valentines day seem to be big ones, lol..)  So, given that some days are more likely for a birthday, I'm assuming the number of 23 is an upperbound.  Has there been a better estimate to this problem? What are homologous chromosomes? My LAN server won't let my cousin join me or me join my cousin. It won't show up on the multiplayer screen - it keeps saying "scanning for LAN servers" but it never finds it. I am at his house and we use different computers but it won't connect. How can I fix this? While doing some flagging (yes, I'm a little bit addicted) I stumbled upon the user . At first the case seemed clear, Spammer, but...and that's a big one, he seems to be a productive member of SU and SO (his account on SF isn't there anymore, but it's still in the Data-Dump of the explorer, not sure what happened).  He does never seem to link against StarWind-Software, except with his name and profile. He also  that he's a developer there...so...at what am I looking here? I can't really place him somewhere. I just bought an Xbox 360 game with wrong region. What should I do? I just bought a  game (Forza 4) for my Xbox 360 (bought in the US). When I tried to play the game, it wouldn't load because of region issue. Is there a legal way to solve this (no crack)? Formula across objects Why is there no permittivity-type constant for gravitation? When I look at electric or magnetic fields, each of them has a constant that defines how a field affects or is affected by a medium. For example, electric fields in vacuum have a permittivity constant $ϵ_0$ embedded in the electric field expression of a point charge: $E = q/4π ϵ_0r^2$. However, if I put this point charge in some dielectric that has a different  permittivity constant $ϵ$, the value of the electric field changes. On a similar note, magnetic fields behave very similar but have the permeability constant $μ_0$ instead.   From my understanding, I believe that this is not the case for gravitational fields since the universal gravitational constant $G$ is consider to be a fundamental constant. So I am assuming that even though gravitational fields do operate in different types of mediums, this somehow doesn’t affect the gravitational field value. My question is why is this the case, that is, why isn’t there a permittivity-type constant for gravitation? Inheriting from a UserControl in WPF I am fairly new to WPF and I am having a problem with inheriting from a user control.  I created a User Control and now I need to inherit from that control and add some more functionality.  Has anyone does this sort of thing before?  Any help would be greatly appreciated.  Thank you I am curious if it is possible to have a VPS that has data on it that is not readable by the hosting provider, but is still usable on the VPS.  Obviously there are some things that you could do to prevent them from reading anything...   You could change all the passwords, including root. But then, they could still use some alternate boot to reset the password, or they could just mount the disk another way. So, you could encrypt the disk or at least some of the contents on the disk. But then it seems that if you decrypted the content, they could still "peer in" to see what you were doing at the console, because after all, the virtualization platform should allow this. And even if you could stop that, it seems they could just read the RAM of the VPS directly.   Of course, the VPS can store data on it and as long as the key is not on the VPS and the data is never decrypted there, then the host cannot get the data.  But it seems to me that if any point the data on the VPS is decrypted...for use on the VPS...then the hosting provider can get the data.  So, my two questions are:   Is this correct? Is it true that there is no way to 100% secure data on a VPS from a host from seeing it, while keeping it accessable by the VPS? If it is possible to make it 100% secure, then how? If it is not possible, then what is the closest you can get to hiding data from the web host? biblatex: Cite commands with the same author prints authors name twice I'm using biblatex 1.4b, and Biber 0.9. However, I think the problem is with my settings, not with the versions of those programs.  I'm trying to cite a single author, and produce their name, followed by the dates of two papers, using the authoryear-ibid style, and natbib compatability. Here's my biblatex preamble:  \usepackage[style=authoryear-ibid,natbib=true,backend=biber]{biblatex}   What I want is for the following command: The works by \citet{Blau81, Blau88} ... to produce: The works by Blau (1981; 1988) ... however, what I'm getting is: The works by Blau (1981); Blau (1988)...  I know that biblatex knows they're the same author, since in the bibliography they fall under the same name (i.e. after the first instance of the name, there's just a dash instead of the name)  What am I doing wrong here? I've also tried it with the commands \cite and \cites, and neither produces what I want.   (crossposted with comp.text.tex) Clear up confusion about the meaning of entropy So I though, and was told, that entropy is the amount of disorder in a system. Specifically the example of heat flow and it flows to maximize entropy. To me this seemed odd. This seemed more ordered to me, from two things (hot and cold) to one thing (lukewarm).   So as I have read today on wikipedia        In recent years the long-standing use of term "disorder" to discuss   entropy has met with some criticism.[20][21][22]      When considered at a microscopic level, the term disorder may quite   correctly suggest an increased range of accessible possibilities; but   this may result in confusion because, at the macroscopic level of   everyday perception, a higher entropy state may appear more homogenous   (more even, or more smoothly mixed) apparently in diametric opposition   to its description as being "more disordered". Thus for example there   may be dissonance at equilibrium being equated with "perfect internal   disorder"; or the mixing of milk in coffee being described as a   transition from an ordered state to a disordered state.      It has to be stressed, therefore, that "disorder", as used in a   thermodynamic sense, relates to a full microscopic description of the   system, rather than its apparent macroscopic properties. Many popular   chemistry textbooks in recent editions increasingly have tended to   instead present entropy through the idea of energy dispersal, which is   a dominant contribution to entropy in most everyday situations.    So what this is saying basically many people mistake the macroscopic outcome of entropy in the same manner as I have above. Seemingly two dissimilar things combining and becoming one.   They say to understand it microscopically. But this still is odd to me, take for example the particles in a box, when they come to thermal equilibrium they are said to have the most entropy they can within their system (from wiki again)     Entropy – in thermodynamics, a parameter representing the state of   disorder of a system at the atomic, ionic, or molecular level; the   greater the disorder the higher the entropy.[7]   What is disorder in the molecular sense I guess is my question. When the particles all come to thermal equilibrium they all seem to be uniform, which I equate with ordering.  I feel disorder in the molecular sense would be fast moving particles, making a mind boggling amount of interactions with each other. Such as in a hot gas, hot gas seems super disordered, but put cold gas in thermal contact with it, let them come to thermal equilibrium, and now they are more ordered!? SubView slide in animation, iphone How old is SpongeBob? How old is SpongeBob? I don't mean how old is the series, I mean what is SpongeBob's age if he were a real person. I ask because his age seems to be ambiguous in the show.   He has a full-time job He has his own house He is always referred to as a kid (especially in the movie) He acts childish and immature In one episode, he was pretending to be an adult/grown-up Alpha and Omega limit sets (dynamical systems) What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$:  I guess I don't really know how to get started, Could I have some guidance? So I was told to calculate the eigenvalues (fine) and eigenvectors (I kind of forgot how to do that...)  I know that the $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the $α$-limit sets of points are the points approached as t goes to negative infinity.   $    A=   \left[ {\begin{array}{cc}    -4 &amp; -2 \\    3 &amp; 11 \\   \end{array} } \right] $   $λ_1=-10$ $λ_2=-5$  Since both eigenvalue are negative, this is a stable node, and the $ω$-limit set=0.  for $λ_1=-10$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    3 \\   \end{array} } \right] $  for $λ_2=-5$:  eigenvector $    =   \left[ {\begin{array}{cc}   2 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    1 &amp; 1 \\    3 &amp; 1 \\   \end{array} } \right] $   $λ_1=4$ $λ_2=-2$  Since one eigenvalue is positive and one is negative, it is not stable, so I can't use the same reasoning that I used above to find the $ω$-limit set... so I am not sure how to continue  for $λ_1=4$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    1 \\   \end{array} } \right] $  for $λ_2=-2$:  eigenvector $    =   \left[ {\begin{array}{cc}   -1 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    4 &amp; 5 \\    -5 &amp; -2 \\   \end{array} } \right] $   $λ_1=1+4i$ $λ_2=1-4i$  for $λ_1=1+4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5-(4i)/5 \\    1 \\   \end{array} } \right] $  for $λ_1=1-4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5+(4i)/5 \\    1 \\   \end{array} } \right]$ How to get the current user in ASP.NET MVC Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$ If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then  $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation that you can find at the  top of page 311 in the 10th edition of Introduction to Probability Models by Sheldon Ross.   Is there a slick or obvious explanation for this identity?   This  question is sort of similar to ; clearly algebra is not my strong suit! Is there an MD5 Fixed Point where md5(x) == x? Is there a fixed point in the MD5 transformation, i.e. does there exist x such that md5(x) == x? There is this puzzling thing that is called : paradoxically, warm (35°C) water freezes faster than cold (5°C) water.  As a physisist, I've been asked about it several times already. And I have no definite answer to that, apart from the standard: "there are many things that can influence it".  So, does anyone knows about the status or progress on that effect? Any recent reviews, publications or other references? I have connected speakers to the back panel of my computer at work and I would like to have them always enabled (I can switch them off on the speaker casing). However when I connect headphones on the front panel system automatically says that the rear output is unplugged.  I found a solution to disable auto detection with Alsamixer and it works almost ok except the fact that after system restart I need to reconfigure the settings again... How can I set this once and forget about it? There are a variety of characters that are not legally encodeable in XML 1.0, e.g. U+0007 ('bell') and U+001B ('escape'). Most of the interesting ones are non-whitespace 'control' characters.    It's clear from (e.g.)  and others that it's  -- but can anyone illuminate me as to why the XML spec forbids these characters?   It seems like it could have been required that they be encoded in escapes, e.g. as &amp;#x0007; and &amp;#x001B; respectively, but perhaps there's a practical reason that the characters were forbidden rather than required to be escaped?  Answerers have suggested that there is some motivation towards avoiding transmission control characters, but Unicode includes many other control-like characters (consider U+200C "zero width non joiner"). I recognize there may be no good reason for this behavior, but I would still like to understand it better.  It's particularly frustrating because when those character values appear in other encodings data formats, I end up "double-escaping" new XML documents that need to encode this.