I was trying to come up with an interesting example (not like $(-1)^n$ or cyclic like $1,2,3, 1,2,3, 1,2,3, \ldots$) of a bounded not converging sequence with an explicit convergent subsequence.   I though of something like $x_n := \sin(n)$. Since $\sin(k\pi) = 0$ for every $k \in \mathbb{Z}$ I thought that the closer the argument of the sine was to a multiple of $\pi$, the closer sine of that argument should be to zero (continuity...). Therefore, the sequence $x_n := \sin(a_n \cdot 10^{n - 1})$, where $a_n$ are the first $n$ digits of $\pi$ should be converging to $0$, right? If yes, how can I show it?
Define a sequence $(a_n)_{n = 1}^{\infty}$ of real numbers by $a_n = \sin(n)$. This sequence is bounded (by $\pm1$), and so by the , there exists a convergent subsequence.  My question is this: Is there any way to construct an explicit convergent subsequence?  Naïvely, I tried considering the subsequence $$\sin(3), \,\sin(31), \,\sin(314), \,\sin(3141),\, \sin(31415),\, \dots$$ hoping it would converge to $0$, but I was unable to prove this (and I think it's probably not even true).
With all the new sites up and running it gets more and more difficult for users to really know the scope of a site and thus a lot of questions a migrated to places where they are (probably even more) off-topic, see e.g. . So I request the opposite of :  Allow migration if you have enough rep on both the source and the target site (the target reputation required could be a bit lower, say 5001)  That way questions could only be migrated by users familiar with the scope of both sites who can really estimate where the question is better. Maybe those who don't have the rep on the target site could still be able to cast a flagging-only-but-not-counting-as-vote vote...  1) Courtesy of Every partially defined isometry can be extended to a isometry I know that the following theorem holds true:  Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual euclidean distance). Suppose that $0 \in S$ and $f(0)=0$. Then $f$ can be extended to an orthogonal transformation of $\mathbb R^n$.  I have two questions:   Can someone give me a reference of a proof of this theorem? I believe that the previous theorem holds true also if we change $\mathbb R^n$ with $\mathbb C^n$ and "orthogonal" with "unitary", with nearly the same proof. Does it holds true also in the $ \ell^2$ space? With the same proof? How many connected components does $\mathrm{GL}_n(\mathbb R)$ have? Help with footer always to bottom . It is a mathematical expression/sequence/set of numbers with the arrow exactly under any one of the numbers (with or without n= 0). How to do it in latex? Any of the up arrow types (single line or double line) shown in the figure is okay. What's the right way to represent phone numbers? Are there functions or packages in Mathematica to sort subsets in lex order, colex, revlex, etc.? Using ls to find files that end in a character, ignoring extension I am supposed to use ls to find files that end in a certain letter, but it does not matter if the file has an extension or not.  For example, I want it to do this  &gt; ls test test.txt test.ascii other.txt  &gt; ls [something] test test.txt test.ascii  So that I can find files that end with a 't', but it doesn't include the file where the extension ends in t  Edit: I am supposed to assume that the only period characters in the filename will be for the extension and there will be no others Can I travel to Kuwait on a visa with old passport number? I have a Kuwait visa, obtained from a hotel in Kuwait, with my old passport number. But as the validity of the present passport is expiring within 6 months, I got a new passport with a new passport number. Can I travel with the same visa or do I have to obtain a new visa with my new passport number? "Misaligned partition" - Should I do repartition (how?) Movie/TV series with a scene of spiky, black goo growing out of a lab container? There was a scene from a sci-fi horror video, not animated, don't know if it's a movie or TV series, where there was this black goo in a glass container and some scientists were looking at it under a microscope.   The pitch black goo grew and grew so fast, eventually breaking out of the inner box and the scientists that were inside the "inner lab" were begging to get out but the scientists outside that inner lab refused to open the door. The black goo was spiky (had spikes).  It's not any of the "Alien" movies, "The Blob", "Life", or "The Thing". It's not vintage. It was made after 1990, I'm pretty sure. How can I clear event subscriptions in C#? Take the following C# class:  c1 {  event EventHandler someEvent; }   If there are a lot of subscriptions to c1's someEvent event and I want to clear them all, what is the best way to achieve this? Also consider that subscriptions to this event could be/are lambdas/anonymous delegates.  Currently my solution is to add a ResetSubscriptions() method to c1 that sets someEvent to null. I don't know if this has any unseen consequences. tmux: shared session, one user in a pane, another in another pane, two different cursors I've decided to try tmux: have been reading the docs and googling around, trying to find a way to have two users sharing a session, each with a different cursor.  However, giving 777 permissions to the socket, or creating a group, chgrping the socket and adding both users to it, seems to let that same socket be used to share a session with only one cursor: both users can write, but always in the same cursor position.  Right now both users are in the same home server over ssh, and the idea is to be able to have:   A terminal in a, let's say, left pane, where I can type commands Another terminal in a right pane, where I can see another user typing commands  in his own terminal The same thing for the other user   What I'm doing at the moment is using two sessions (not shared) and a script -f and tail -f combination that kinda works for reading each other's key strokes, but I reckon there is probably some way of doing this using tmux sharing capabilities.  Is there a way to get this idea working with write support in each other's terminal?  What is the better way to do this? How can I check if a word is contained in another string using PHP? Is there a ".bashrc" equivalent file read by all shells? Is ~/.bashrc  the only place to specify user specific environment variables, aliases, modifications to PATH variable, etc?  I ask because it seems that ~/.bashrc seems to be bash-only, but other shells exist too… How to detect if a function is called as constructor? Given a function:  function x(arg) { return 30; }   You can call it two ways:  result = x(4); result = new x(4);   The first returns 30, the second returns an object.  How can you detect which way the function was called inside the function itself?  Whatever your solution is, it must work with the following invocation as well:  var Z = new x();  Z.lolol = x;  Z.lolol();   All the solutions currently think the Z.lolol() is calling it as a constructor. What do you call the frequent conjunction {juxtaposition} of two words? The statement that I want to make is:     "Impostor syndrome" ... why does this (conjunction of words) sneak into every text I read these days?   What is the right or best expression for "conjunction of words"? In the statement I don't want to refer to the syndrome itself, but rather to the expression. If it was one word instead of two I would want to say     ... why does this word sneak into every text I read these days.   However, writing     ________ why do these two words sneak into every text I read these days.   would miss the point, because it's not these two words that sneak into every text at various location but rather their conjunction. Regarding the `book`, `report`, and `article` document classes: what are the main differences? Comment font sizes are inconsistent on Android What is the optimum pattern to place crops in Minecraft? What factors affect soil hydration and crop growth time? What is the most efficient layout of dirt and water to grow crops?