Show that the set of accumulation points of the sequence $a_n = \sin{n}$ is the closed interval $[-1, 1]$ I'm learning about Real Analysis, specifically accumulation points of a sequence, and need some help with this problem:      Show that the set of accumulation points of the sequence $a_n = \sin{n}$ is the closed interval $[-1, 1]$.     My thoughts:   Obviously we have $-1 \leq \sin{n} \leq 1 \forall n \in \mathbb{N}$.   The set $a_n$ will be of the form  $a_n = \{\sin1, \sin2, \sin3, ...\}$.   Given $l \in [-1, 1]$ we want to show that there exists a subsequence  $\{a_{n_k}\}$ such that $a_{n_k} \rightarrow l$ as $k \rightarrow \infty$.    This problem is different from the problems I'm used to solve in the sense that instead of finding the accumulation points of a sequence I need to prove that the accumulation points of a sequence is a closed interval and I don't know how to approach this.
$\sin(n)$ subsequence limits set We were given a challenge by our calculus professor and I've been stuck on it for a while now.   Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).  What would be a good way to start?
Let's assume we have a table where it has two columns, one an ID and one an hstore column, which stores key/value pairs.  You could query that column as such:  SELECT name, attributes-&gt;'device' as device FROM products WHERE attributes-&gt;'edition'= 'ebook'   Can you create a view for this table where the keys/values appear as columns in the view instead of all in the hstore column?  Example: I have an hstore key where it might have the key of amenity.  I want to create a view showing the the ID and the amenity values in a new view. Multiple arguments in shebang I am wondering whether there is a general way of passing multiple options to an executable via the shebang line (#!).   I use NixOS, and the first part of the shebang in any script I write is usually /usr/bin/env. The problem I encounter then is that everything that comes after is interpreted as a single file or directory by the system.  Suppose, for example, that I want to write a script to be executed by bash in posix mode. The naive way of writing the shebang would be:  #!/usr/bin/env bash --posix   but trying to execute the resulting script produces the following error:  /usr/bin/env: ‘bash --posix’: No such file or directory   I am aware of , but I was wondering whether there was a more general and cleaner solution.    EDIT: I know that for  scripts, there is a way to achieve what I want, documented in  of the manual:   #!/usr/bin/env sh  exec guile -l fact -e '(@ (fac) main)' -s "$0" "$@"  !#   The trick, here, is that the second line (starting with exec) is interpreted as code by sh but, being in the #! ... !# block, as a comment, and thus ignored, by the Guile interpreter.  Would it not be possible to generalize this method to any interpreter?     Second EDIT: After playing around a little bit, it seems that, for interpreters that can read their input from stdin, the following method would work:  #!/usr/bin/env sh sed '1,2d' "$0" | bash --verbose --posix /dev/stdin; exit;   It's probably not optimal, though, as the sh process lives until the interpreter has finished its job. Any feedback or suggestion would be appreciated. Why can not upgrade texlive2016 to texlive2017 on windows I have texlive 2016 installed on my windows OS, to use texlive 2017 you need to download it (over then 2.5GO), I think this is a tedious operation every time you  want to pass from an old texlive to a recent one.  My question is is there a posibility just to upgrade texlive 2016 to texlive 2017without removing, downloading, and installinig  the entire texlive? How can I convert my Java program to an .exe file? How do I fix gaps around my lean-to greenhouse? Clean way of building simple taxonomy browser of arbitrary depth How to use HttpWebRequest (.NET) asynchronously? How can I use HttpWebRequest (.NET, C#) asynchronously? How to format/partition SD card? How can I format or partition SD card in Kubuntu?  Is there an application with GUI for this? Suppose all the Associated Primes are Minimal. Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.  1) Suppose $\operatorname{Ass}(R/I)$ are all minimal, then what can we say about $R/I$?  One conclusion is that $I$ has a unique primary decomposition. If $R/I$ is CM, then $\operatorname{Ass}(R/I)$ are all minimal, but I don't think the converse is true. Can someone provide a counterexample?  2) Suppose $\operatorname{Ass}(R/I)$ are all minimal and have the same height. I still dont think we can conclude that $R/I$ is CM. Can someone provide a counterexample? Theorem 2.1.6 of Bruns and Herzog, CM Rings, gives a result in this direction but it needs additional hypothesis. It says that if every ideal $I$ is generated by height $I$ elements and $\operatorname{Ass}(R/I)$ are all minimal, then $R$ is CM.  It would be nice if someone can say something nice about the ring in the special situation described in 1) and 2).  Thanks. mysql select distinct rows into a comma delimited list column Had it not been vs If it were not for I've just seen the following sentence on a band's biography:     Had it not been for a studio theft, the band might never have made that song.   Then I was wondering if I can change this sentence to:     If it were not for a studio theft, the band might never have made that song.   Is the second sentence correct? Are they interchangeable?   Thanks in advance. Determining relationship (e.g. number of removes) with cousin? How do I figure out if someone is a first cousin, second cousin, cousin once-removed, etc? Gravitational potential at a point in a gravitational field is defined as work done per unit mass in bringing a small test mass from infinity to the point.  I know that:   By integration, we find that gravitation potential is negative. The maximum value of gravitational potential which is at infinity distance is 0, therefore minimum value should be negative.   But, what I do not understand is force is in the same direction as displacement so why work done is not positive? Why are only certain shutter speeds or apertures available? Cardinality of the Irrationals Possible Duplicate:          We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real numbers, is uncountable. What can you say, then, about the cardinality of the irrational numbers?  I would say that the cardinality of irrational numbers is uncountable. That would be because the set $\mathbb{Q}$ is a subset of $\mathbb{R}$. The irrationals is also a subset of $\mathbb{R}$. Basically, my thought is that the continuum is composed of irrationals and rationals (only two subsets of reals). Because we can prove that the rationals are countable by Cantor's proof, then the remaining subsets of $\mathbb{R}$, the irrationals, has to be uncountable for $\mathbb{R}$ to be uncountable. My concern is how to go about the prove this (rigorous, intro level). I know this subject was discussed for many times, but I still don't get it..  I've read in a book that we can use "I've got" instead of "I have" for possesion and it gave some examples,      I've got an English book.   I've got a red car.   So I understand that it can be used only for possession of physical objects, but I saw it used in sentences like "I've got something to say", is it correct? I hope it's not a stupid question..Thank you very much! Installing File Geodatabase (*.gdb) support in QGIS? I have spent around 2 days to find the way to open GDB (Esri geodatabase) in QGIS (or any other open source software) but still without success.  I have downloaded the newest OSGeo4W installer and tried the setup - express desktop install - all packages, as well as advanced install incl gdal-filegdb.  Can you describe a more detailed procedure, including installation and how to open .gdb in QGIS (OSGeo4W installation)? Where were the Avengers during Iron Man 3? What is a NumberFormatException and how can I fix it? Solve $$u_{xx}-3u_{xt}-4u_{tt}=0$$ where $u(x,0)=x^2$ and $u_t(x,0)=e^x$.  My workings so far: I have factored the differential equation in the following way: $$(\delta_x-4\delta_t)(\delta_x+\delta_t)=0$$ where $\delta_x=\frac{\delta}{\delta x}$ etc. Now if we let $v$ be the solution to $(\delta_x+\delta_t)u$ then we have the following two equations: \begin{eqnarray*} (\delta_x+\delta_t)u=u_x+u_t=v\\ (\delta_x-4\delta_t)v=v_x-4v_t=0 \end{eqnarray*} Now for $v$ we simply find $$v=h(t+4x)$$ where $h$ is an arbitrary function of one variable. Now what remains is find $u$ such that $$u_x+u_t=h(t+4x)$$ I am stuck here, I thought about making a change of variables $\zeta=x+t$ and $\eta=x-t$ and thus using the product rule to show that $u_x=u_{\zeta}+u_{\eta}$ and $u_t=u_{\zeta}-u_{\eta}$ and thus $u_x+u_t=2u_{\zeta}$ and we need to solve $$u_{\zeta}=h(t+4x)$$ (I left out the factor 2 because $h$ is an arbitrary function). Do I simply integrate now and conclude $$u=f(\zeta)h(t+4x)+g(\eta)=f(x+t)h(t+4x)+g(x-t)$$ This seems wrong to me...  Some help would be greatly appreciated!