Let $X$ be a Banach space with norm $\|\cdot\|$. Let $(x_n)$ be a Cauchy sequence in $X$ containing a subsequence $(x_{n_k})$ which converges to some $x \in X$.   Is it true that $(x_n)$ must also converge to $x$?   I think the answer is yes, since $$\|x_n - x\| \leq \|x_n - x_{n_k}\| + \|x_{n_k} - x\| &lt; \epsilon,$$ but I would appreciate any verification/correction.
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.
Awaken monit daemon every few hours for all monitored processes I'm building a small chat. Now I see but in other chats (whatsapp, facebook mobile chat, sms ...) that the arrangement is always the same:  Left: the chat partner. right: myself      Why is that? Getting the class name of an instance? How do I find out a name of class that created an instance of an object in Python if the function I am doing this from is the base class of which the class of the instance has been derived?  Was thinking maybe  might have helped me out here, but it doesn't seem to give me what I want. And short of parsing the __class__ member, I'm not sure how to get at this information. If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$ I'm going over old exam problems and I got stuck on this one. Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be the non-zero zeros of $\mathbb{f}$ in $\mathbb{D}$ counted according to multiplicity. Show that  $$ \sum_{n=1}^\infty \left( 1 - \left|a_n \right|\right)\lt\infty $$  I can understand that $\left|a_n \right|$ goes to $1$ since zeros are isolated, but it doesn't help showing the series is convergent. Any help will be appreciated! How do you reset the sa password? How much meat should I serve per person? Does a controlled mount share its rider's turn? From "Controlling a Mount" on p.198 of the Player's Handbook:     The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.   Does this mean that the rider and mount share a single turn? Electromagnetic black hole? So I was thinking about something for the past while  Consider a large spherical foam-ball with homogeneous density. Where a foam ball is defined as an object that can absorb matter with 0 friction (example: a gravitational well without an object inside). This is a purely theoretical construct.  If the foam-ball has a radius R and a charge Q. What charge must the foam-ball have, such that there is well defined sphere or horizon such that any object with a negative charge (even if it is the slightest bit) must travel faster than C in order to escape the field of the ball.  Aka what charge would turn this into a black-hole for all objects with opposite sign on their charge?  Once again I assume the foam-ball is a single particle, that doesn't repel itself... its just a very large homogenous 'thing' with charge. Edited photos have reverted to originals on iphone In my experimentation, I've used only BJTs as switches (for turning on and off things like LEDs and such) for my MCU outputs. I've been repeatedly told, however, that N-channel enhancement-mode MOSFETs are a better choice for switches (see  and , for examples), but I'm not sure I understand why. I do know that a MOSFET wastes no current on the gate, where a BJT's base does, but this is not an issue for me, as I'm not running on batteries. A MOSFET also requires no resistor in series with the gate, but generally DOES require a pull-down resistor so the gate doesn't float when the MCU is rebooted (right?). No reduction in parts count, then. There doesn't seem to be a great surplus of logic-level MOSFETs that can switch the current that cheap BJTs can (~600-800mA for a 2N2222, for example), and the ones that do exist (TN0702, for example) are hard to find and significantly more expensive. When is a MOSFET more appropriate than a BJT? Why am I continually being told that I should be using MOSFETs? I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this? Make a computer act as a virtual USB device for other equipments I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan It is much more convenient and cleaner to use a single statement like  import java.awt.*;   than to import a bunch of individual classes  import java.awt.Panel; import java.awt.Graphics; import java.awt.Canvas; ...   What is wrong with using a wildcard in the import statement? How to create csv files in controller method from list I want to create csv files from lists and send an email attaching them, how to do that in a controller method? How can you move a document with its version history to a new library? Here's the scenario:  2 document libraries Both contain multiple documents The documents have multiple versions  When I copy a document from library A to Library B using the Content and Structure tool from Site Settings, my versions do not carry over. I'm trying to merge the two document libraries into one, but I want to retain the versions. Do you know of a way to complete this without third party tools or custom code? I've heard suggestions of using Explorer View copy and paste, SharePoint Designer and List Templates, but I'm not sure any of those will carry version info. Modern solution For modern SharePoint experiences, use Move To. Update 4/24/2018: There's a new solution in Office 365. You can now select a file or group of files in the modern experience in OneDrive or SharePoint and select Move to. A file or group of files moved this way will retain version history. Update 6/6/2021: As @KJH mentioned in a comment, you may not see the modern experience unless you . Also, there's more detail now in the Microsoft article . What do ellipsis [...] mean in a list? Suppose A is a 3*3  symmetric matrix s.t.  $$\begin{pmatrix} x &amp; y &amp; 1 \\   \end{pmatrix} A  \begin{pmatrix} x  \\  y\\ 1\end{pmatrix}  = xy -1 $$  let p be the no. of positive eigen value of A and q = rank(A)-p, then p=? q=?  what matrix A should be I can't find?? How can $\pi$ be irrational if it can be calculated using an infinite series of rational numbers? I want a true deep copy. In Java, this was easy, but how do you do it in C#?