Let $(X,d)$ be a metric space and let $S$ be the set of Cauchy sequences in $S$. Define "~" in $X$ by declaring "$\{s_k\}$ ~ $\{t_k\}$" to mean that $d(s_k,t_k) \to 0$ as $k \to \infty$. Let $X'$ denote the set of equivalence classes of $S$ and let $s'$ denote the equivalence class of $s = {s_k}.   I showed that $\phi(s',t') = \lim\limits_{k \to \infty} d(s_k,t_k)$ is a metric on $X'$.   Now I need to show $(X',\phi)$ is complete. Then for $x \in X$, define $x'$ to be the equivalence class of the constant sequence ${x,x,...}$. Show that the function $x \to x'$ is an isometry of X onto a dense subset of X.   I don't even know where to start. I need to show for the second case that for any $y' \in X'$, we can find a sequence that converges to $y'$. Any help please?
Let $(X,d)$ be a metric space and let $S$ be the set of Cauchy sequences in $S$. Define "~" in $S$ by "$\{s_k\}$ ~ $\{t_k\}$" to mean that $d(s_k,t_k) \to 0$ as $k \to \infty$. Let $X'$ denote the set of equivalence classes of $S$ and let $s'$ denote the equivalence class of $s = \{s_k\}$.   I know $\phi(s',t') = \lim\limits_{k \to \infty} d(s_k,t_k)$ is a metric on $X'$.   Show $(X',\phi)$ is complete.  For $x \in X$, define $x'$ to be the equivalence class of the constant sequence ${x,x,...}$. Show that the function $x \to x'$ is an isometry of $X$ onto a dense subset of $X$.   If anyone can share a proof, or steer me in the right direction, maybe tell me what it is that I need to prove specifically, that would be great.
How to add home directory path to be discovered by Unix which command? Invalid SSL certificate when pushing to Git server For a 3D scene in the world coordinates, its View Reference Point $\mathrm{VRP}$ is at $(5,-2,1)$, and a viewer is looking towards point $A=(1,1,1)$. Construct a transform matrix which will map world coordinate points to a right-handed $(UVN)$ viewing space, so that $\mathrm{VRP}$ is the origin, the line joining $\mathrm{VRP}$ to $A$ is the positive $N$ axis, and the view-up Vector is $(0,0,1)$.   To my knowledge, the final answer should be a single transform matrix. To calculate the final matrix, I first find the matrix which translates the coordinate system to the to the origin in homogeneous matrix form:  Translation matrix:  $$ T=         \begin{bmatrix}         1 &amp;0 &amp;0 &amp;-5 \\         0 &amp;1 &amp;0&amp; 2 \\         0 &amp;0 &amp;1 &amp;-1 \\         0 &amp;0 &amp;0 &amp;1 \\         \end{bmatrix} $$  Using the unit vectors of the coordinate axes, the resulting rotation matrix is:  $$ R=         \begin{bmatrix}         -0.6 &amp;-0.8 &amp;0 &amp;0 \\         0 &amp;0 &amp;0 &amp;1 \\         -0.8 &amp;0.6 &amp;0 &amp;0 \\         0 &amp;0 &amp;0 &amp;1 \\         \end{bmatrix} $$ Then the final matrix (the final answer for full marks) would be the rotation matrix multiplied by the translation matrix:  $$ M=         \begin{bmatrix}         -0.6 &amp;-0.8 &amp;0 &amp;1.4 \\         0 &amp;0 &amp;0 &amp;1 \\         -0.8 &amp;0.6 &amp;0 &amp;5.2 \\         0 &amp;0 &amp;0 &amp;1 \\         \end{bmatrix} $$ Is my method and final answer correct?    Following the answer to this question, I have calculated the translation matrix:  $$ T= \begin{pmatrix} 1 &amp; 0 &amp; 0 &amp; -5 \\ 0 &amp; 1 &amp; 0 &amp; 2 \\ 0 &amp; 0 &amp; 1 &amp; -1 \\ 0 &amp; 0 &amp; 0 &amp; 1 \\ \end{pmatrix} $$  I have calculated the rotation matrix:  $$ R= \begin{pmatrix} -0.6 &amp; -0.8 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 1 &amp; 0 \\ -0.8 &amp; 0.6 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 1 \\ \end{pmatrix} $$  But what would the final transformation matrix $M$ be? My lecture slides tell me it would be rotation matrix multiplied by translation matrix, with the last column multiplied by $-1$ except for the $1$. This would equal:  $$ M = R \times T =  \begin{pmatrix} -0.6 &amp; -0.8 &amp; 0 &amp; -1.4 \\ 0 &amp; 0 &amp; 1 &amp; 1 \\ -0.8 &amp; 0.6 &amp; 0 &amp; -5.2 \\ 0 &amp; 0 &amp; 0 &amp; 1 \\ \end{pmatrix} $$  But the answer here just adds them together somehow. Which one is correct? Calling Table with custom iterator CSOM: Identify "columns" created by user rather than built-in When iterating the fields of a list how do you know which columns are yours and which are built-in ?  I know a solution for "classic" SDK using SPField.SourceId  but I can't find one for CSOM.  I thought UserCreated was the way to go for but   So how do you tell apart theirs and mine ? Using a combination of Hidden CanBeDeleted FromBaseType ? Checking for a list of special internalNames ?          ClientContext context = new ClientContext(siteUrl);         context.Credentials = creds;         Web web = context.Web;         var list = web.Lists.GetByTitle(listName);          context.Load(list.Fields);         context.ExecuteQuery();          foreach (Field field in list.Fields)         {         bool thisColumnWasCreatedByUser =                        field .FromBaseType == false &amp;&amp;                     //this.InternalName != "Title" &amp;&amp; // builtin but I'd rather keep it                     field .InternalName != "_dlc_DocId" &amp;&amp; // probably safe                     field .InternalName != "_dlc_DocIdUrl" &amp;&amp;                     field .InternalName != "_dlc_DocIdPersistId" &amp;&amp;                     field .InternalName != "Combine" &amp;&amp; // sucks                     field .InternalName != "RepairDocument" // sucks too         }   EDIT  Mixing suggestions by Viraj Gorajia and Dikesh Gandhi, my current solution is  bool thisColumnWasCreatedByUser =  field .FromBaseType == false &amp;&amp; (!field .SchemaXml.Contains(" SourceID=\"http")) &amp;&amp; // remove "Title", "Combine", "RepairDocument"  (!field .EntityPropertyName.StartsWith("OData__")) // field .Sealed == true could be working too but I don't know if it could exclude valid results Redirect the output of a command in `time command` I am trying to time something using:  /usr/bin/time myCommand   However, since /usr/bin/time writes to stderr, if myCommand also writes to stderr, I will get more than just time's output on the stream. What I want to do is, redirect all of myCommand's output to /dev/null, but still write time's output to stderr. Using an example myCommand that writes to stderr of ls /nofile, we see that (obviously) there is no output at all with the following:  $ /usr/bin/time ls /nofile 2&gt; /dev/null $   Without any redirection, we see both the output from ls (to stderr) and the output from time (also to stderr):  $ /usr/bin/time ls /nofile ls: cannot access /nofile: No such file or directory 0.00user 0.00system 0:00.00elapsed 0%CPU (0avgtext+0avgdata 3776maxresident)k 0inputs+0outputs (0major+278minor)pagefaults 0swaps   What I want is something that simply produces:  $ /usr/bin/time ls /nofile &gt; RedirectThatImAskingFor 0.00user 0.00system 0:00.00elapsed 0%CPU (0avgtext+0avgdata 3776maxresident)k 0inputs+0outputs (0major+278minor)pagefaults 0swaps   Any ideas? When we refer to a house that belongs to a family, we say "family's house". Pluralizing family gives us "families". Referring to the houses of several families, we say "families' houses". Forming the plural possessive in such a case is rather simple.  I encountered a more complex use of this recently - referring to a single house owned by a single family in a set of houses owned by a set of families.  We can say "the families' houses" to refer to all of the houses owned by all of the families.  We can say "one of the families' houses" to refer to one of the houses owned by one or more of the families. In this context, "one of" applies to "the families' houses".  If we were instead to apply "one of" to "the families", and want to refer to the one house owned by "one of the families", how would this be written?  My first though was "one of the families's house", taking "one of the families" as a single noun and appending 's to it. This looks (and sounds) a bit strange, though. "One of the families' house" and "one of the family's house"/"one of the family's houses" seem wrong to me, and I can't really determine the correct way to say/write this.  Is there a definitively correct way that this should be written?  EDIT: To clarify, although I think I made it clear, in this context, there are several families, and each owns one house. The goal of the sentence is to refer to one of those houses without stating which specific house. Example: "The players and their families want to celebrate their team's victory; this will likely entail a party at one of the families's house". How can I send an email using PHP? What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE? Since when did kidnapping come to include adults too? As per , the word 'kidnap' originated to denote nabbing away of a child. When and how did kidnap come to denote nabbing of adults?  Update: Just found a link to a  that mentions kidnapping of souls. Now not sure how correct the above link is. There's already a question like this  so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.   Is there a way to derive  from the Lorentz' Force law or just from Maxwell's Equations?   The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.   Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?  I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.   In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow? Any open subset of $\Bbb R$ is a countable union of disjoint open intervals When on the road as a tourist, I often do not carry a reflector of flash. Now I am at a location with very strong sunlight and I want to take a photo of a companion in front of a nice scenery or building, the classic "we have been here" photo.  With very strong sunlight however, specially at a slight angle, I either get it into the person's eyes and they squint like crazy, or they are in the shade and their whole face is black.  Please note that I do not have aspirations to take a professional photo in such circumstances, but at least something acceptable.  Is there an easy trick to deal with that? How do I get rid of my bounty? I accidentally killed a noble, now every guard wants to kill me. Is there any way I can get rid of the bounty? Is there a way to override class variables in Java? Share wireless Internet connection through ethernet I want to share the wireless Internet connection on my desktop with my old laptop, through ethernet.  I have ticked the option "Make available to others" on both connections, on my desktop, but my laptop doesn't connect. Am I doing something wrong or is it a bug? Let $\mathbb S = \varnothing$.      Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$      Consider any $x \in \mathbb U$.      Then as $\mathbb S = \varnothing$, it follows that: $\forall X \in \mathbb S: x \in X$ from the definition of vacuous truth.      It follows directly that: $\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$      That is: $\bigcap \mathbb S = \mathbb U$. ()   Proofwiki uses the above "proof" to "prove" that intersection of the empty set is the whole universe.   My question is, is the use of vacuous truth really allowed in axiomatic set theory, like ZFC? I don't see how the use of vacuous truth is justified.   The next problem I can think of is that we cannot really "define" the elements of empty set (to my knowledge, there is no element in empty set) so how can we then prove as the above proof did? This seems to contradict the use of vacuous truth.  And of course, there is issue of using the whole universe as a set, and I don't think this is allowed.... (Maybe proof above is using a different axiomatic set theory, as I am using ZF-minded thoughts...) Can you set a border opacity in CSS? "[" as first character in table row Whenever I try to compile a table that has a "[" as first character in a row it gives me the error:  Missing number, treated as zero. Illegal unit of measure   Example:  \begin{table}[htb]     \centering     \begin{tabular}{r|r}         [some text ] &amp; ... \\         [some more text ] &amp; ...     \end{tabular}     \caption{...}     \label{tab:...} \end{table}   How can I print "[" as first character in a row? Hello,   I create an app using the Promoted Links in Sharepoint 2013. I chose to do this because the feature has the awesome hover effect and is easy to customize the image.  The issue is that I have been trying to figure out how to change the tile sizes. They are awesome the way they are, however I just need them smaller. I still want the hover effect as well and for the text to match the same ratio as the tile. I have tried web zones, I have done each ID but with no luck. I have attached a sample image of the tiles I'm referring to. Please help.