Is it true that $x_n$ is a Cauchy sequence in some Hilbert space if and only if:  for all $m \in \mathbb{N}$, for all $\epsilon &gt; 0$, there exists $N=N(m) \in \mathbb{N}$ (so the $N$ can depend on $m$) such that $$\lVert x_{n+m}-x_n \rVert \leq \epsilon \quad\text{ if $n \geq N(m)$}.$$  Is that correct?
Is this statement true?      In a metric spase $(E,d)$, a sequence $(x_n)$ is Cauchy if and only if $  \forall k\in \mathbb{N}, \lim_{n\rightarrow+\infty}d(x_{n+k},x_n)=0$   I proved that $\Rightarrow$ is true,indeed if $(x_n)$ is Cauchy then for all $\varepsilon&gt;0,\exists n_0\in\mathbb{N}, \forall p,q\geq n, d(x_p,x_q)&lt;\varepsilon$ it still right for all $ n\geq n_0$ and $m=n+k&gt;n\geq n_0$   But I don't know if $\Leftarrow$ is true ?
What is the term for the grandson of my aunt? Why is there no 32 bit ISO for the new Ubuntu 18.04? Why is there no 32 bit ISO for the new Ubuntu 18.04? Proof of an elementary property of Projection Operators I'm asked to show the following:  Let $X$ be a linear space, and let $P : X \rightarrow X$ be a projection operator. Restricted to the linear space $range(P)$, the projection $P$ is the identity operator, that is, $Px = x$ for all $x \in range(P)$.  If anyone could offer a hint for how to show this. I'm rusty with linear algebra so don't really know what I'm supposed to bring in outside of what's specified in the lemma/definition of projection. What is so special about the range of P that would make this so? Is there a ZIP code I can enter when paying-at-the-pump in the USA with a foreign credit card? Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$. Find the image of (1,1,1) by T Question: Find the image of (1,1,1) by T  $$T(x,y,z) = (y,6y,x+y+z)$$   I found the standard matrix to be:   \begin{bmatrix}0&amp;1&amp;0\\0&amp;6&amp;0\\1&amp;1&amp;1\end{bmatrix}  I'm confused on how to approach this question.   In order to find the image is it possible to inverse the values of it's standard matrix?  Would I able to use the identity matrix of 3 and inverse the values of that to find the image?    Any help will be greatly appreciated. I'm writing a thesis which is an aggregate of separate articles. I have the source code for the articles and I'm using \include to include them in the main document of the thesis. So in my main.tex (thesis document), I have  \include{article1} some pages in between the two articles \include{article2}   The problem that I'm getting is with the Appendix sections. In article 1, I have at the end:  \appendix \numberwithin{equation}{section} \section{Coefficients}\label{sec:Coeffic} \subsection{Good coefficients} then comes the first line of appendix   This gives in article 1 (when  compiled separately with the article class):     A. Coefficients      A.1. Good coefficients   and equation numbers in the Appendix A as (A.1), (A.2)  which is ok and what is expected.  When I put article 1 within main.tex, I use \include command. Also, I removed the document class for article 1 and put \chapter instead.  But when I compile main.tex (the thesis document), I get the appendix sections as:     .1 Coefficients      .1.1 Good coefficients   and the equation numbers in the appendix A as (.1.1), (.1.2) etc  My article 1 is treated as a chapter in the main thesis document (main.tex) and the title and section numbers, equation numbers are preceded by A in main.tex. This is what I want.  Maybe it would be ok to get the appendix in my main.tex as     A.A. Coefficients      A.A.1 Good coefficients   and the equation numbers in the Appendix as (A.A.1), (A.A.2) etc  to denote that it refers to appendix A of paper A. First A is for the first article and second A is for first appendix. How to get this? How can I convert JPG into PDF easily? Is there any way to batch convert .jpg files into single .pdf files? Check if partition is encrypted I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? Using something like this:  try:    # Something... except Exception as excep:    logger = logging.getLogger("component")    logger.warning("something raised an exception: " + excep)    logger.info("something raised an exception: " + excep)   I would rather not have it on the error-level cause in my special case it is not an error. Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]:     Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field  and let $c \in  F$. Then $x^p - c$ is irreducible in $F[x]$ if and only if $x^p - c$   has no root  in $F$. [Hint: consider two cases: char$(F) = p$ and   char$(F)$ different of $p$.]   I have attempted this a lot. Anyone has an answer? My wireless/WiFi connection does not work. What information is needed to diagnose the issue? Your situation   You have successfully installed Ubuntu.  You have just downloaded and booted Ubuntu live media.  The latest LTS (see also ) or latest non-LTS release are preferred. See the list of  that are currently supported.) You upgraded your Ubuntu installation to the latest release that the software updater offered you. WiFi worked before, but not now on the new release. You  to new hardware.   Your problem   The wireless of your laptop or desktop is not working. You tried switching the wireless switch off and on and you tried rebooting several times, but you don't see any WiFi access points. You can see your wireless access point, but you cannot establish a connection. You want to analyze the problem, but you don't know where to start or what information you can provide.   This question and answer was written to give people the wireless script that will provide the information needed in most cases to diagnose their wireless issue's.   Related questions How to specifically use ntfsundelete to recover directory This is a followup on my original post  which may be deleted in the future.  In the process of backing up my audio books, I "permanently" removed a folder (with subfolders) that I had already exclusively moved to that drive. It's path is /media/gratis/Seagate Expansion Drive/Audiobooks/Jason Dark. The sole folder in that should be called John Sinclair; but I'm not sure about that.  Which tool offers the best chances to recover those files? I haven't written plenty on that drive afterwards; so I hope for the best. It's not that other methods failed; I just cant get my head around what to do and I want to be cautious. It's a NTFS Drive.  According to testdisk it has an "EFI GPT" Partition table.  Disk /dev/sdb - 5000 GB / 4657 GiB - CHS 608001 255 63, sector size=512 - Seagate Expansion Desk, FW:9401   Information According to "Disks" (Ubuntu Standard Software):  NTFS; GUID Partition Table; [...] Partition Type: Microsoft Reserved   Attempt 1: First off, ntfsprogs seems to no longer be in repos (those mentioned by another user); but it is already installed on my machine so no problems there. I always get the error:     Access is denied because the NTFS volume is already exclusively opened. The volume may be already mounted, or another software may use it which could be identified for example by the help of the 'fuser' command. You can use force option to avoid this check, but this is not recommended and may lead to data corruption.   Although I only just plugged it in and even quit Nautilus by force (nautilus -q).   Therefore, I ran sudo ntfsundelete /dev/sdb2 -f. That gave me a list of a bunch of files, 205013 to be exact. Now, what I want is the content of the directory .../Audiobooks/Jason\ Dark/ to be restored. What do I have to do? It's contents are .m4b &amp; .opus files and folders, of course. What does "overfull hbox" mean? (Why is there a black mark at the end of a line?) I often see output from TeX with the warning overfull hbox, badness 10000. What does this message mean? Could normal users have an easier way to flag for migration? I am running Ubuntu 12.04 on an IBM ThnkPad T42.  I just installed Gnome and Cinnamon but I am still using Unity.  The only problem is that in dash when I search for any application they don't appear.  I don't see recent applications either.  I fixed it by running unity --reset but the next time I turned on my computer it was broken again Does a single photon's energy depend on frequency? (photoelectric effect) How can I communicate with a Unix domain socket via the shell on Debian Squeeze? I’m running a Debian Squeeze web server. I’ve installed memcached on it, and configured memcached to listen on a Unix domain socket (at /tmp/memcached.sock), as it only needs to receive messages from the website, which lives on the same server.  It seems to be working fine, but I’d also like to communicate with memcached via the shell, to check that it’s doing what I think it’s doing.  memcached accepts messages via a simple ASCII protocol (if I understand correctly). If it was listening on TCP/IP, I could send messages to it via e.g. nc:  $ echo "stats settings" | nc localhost 11211   But I can’t figure out how to send that text to the domain socket instead.  On my laptop (which runs OS X Lion), both nc and telnet have options (-U and -u respectively) to use domain sockets. However, on my Debian Squeeze web server, these options aren’t present. Orphaned tag wiki edits have broken activity entries