I am supposed to show that $\sin(n)$ does not converge by constructing two subsequences: one subsequence contains terms of $\sin(n)$ that are between $1/2$ and $1$, and the other subsequence contains terms of $\sin(n)$ that are between $-1/2$ and $-1$. But how can I do this? Which $k(n)$--for subsequence $x_{k(n)}$--should I choose?
Show that $a_n=\sin(n)$ does not converge     My idea:  Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$  So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, $\lim_{n\to\infty} a_{n_l}=-1$  So two inifinite subsequence converge to different limits thus the sequence $a_n$ doesn't converge.   Is that correct ?  Edit:   The contrapositive of the defintion of a limit of a sequence:   $\exists\epsilon&gt;0 : \forall n\in N : \exists n&gt;N \Rightarrow |x_n-L|&gt;\epsilon$  Take $\epsilon =1$ and we know that $\sin(n)$ is bounded so lets take it's supermum: 1  $|1-1|&gt;\epsilon=1\Rightarrow 0&gt;1 \Rightarrow$ Contradiction.
Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{1}\not\in\mathfrak m_iM_{{\mathfrak m}_i}$ for every $i=1,\dots,n$.    I cannot prove that such an element there exists. I was trying to prove it by induction on $n$. If $n=1$ it is true by Nakayama, so suppose it is true for $n-1$. Then for every $i$ I can find an $x_i\in M$ such that $\frac{x_i}{1}\not\in \mathfrak m_jM_{\mathfrak m_j}$ for every $j\neq i$. If $\frac{x_i}{1}\not\in \mathfrak m_iM_{\mathfrak m_i}$ for some $i$ we are done, so suppose $\frac{x_i}{1}\in\ m_iM_{m_i}$ for every $i$. I don't know how to go on, could you help me? It would seem that far-away stars are at such a distance that I should be able to take a step to the side and not have the star's photons hit my eye. How do stars release so many photons to fill in such great angular distances? What is the difference between max, min and sup, inf? rake db:migrate doesn't seem to work in production Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$? Create an instance of a class from a string Simplest way to prevent iTunes from automatically launching when iPhone/iPad is connected When I connect my iPad or iPhone to my Mac iTunes automatically launches. What is the simplest way to prevent this from happening? Blender 2.78 with Windows 10 lags when interacting with the UI Update  This has been confirmed resolved in a later version 376.33 and possibly the update prior.     Okay, I recently did a computer update. I used to run Blender 2.77 on a Dell desktop (5 GB RAM, 2.7 GHZ, Windows 8, build in graphics card). Since that computer wasn't fast enough for my preference, I saved up and got a new gaming laptop - the Asus ROG GL552 (Windows 10, 16 GB DDR4 RAM, Nvidia GeForce GTX 960M with 4 GB DDR5 RAM, 2.6-3.5 Ghz, and quad-core intel processor). That laptop has been known to have outdated drivers, so when I got it, I updated it to up to date Windows 10 drivers.  As soon as I started Blender, I immediately noticed its speed. Even doing simple things, like clicking buttons, or rotating the view with the middle mouse button, took too long to register in the program. I tried changing some User Preference System settings, but that didn't really help much. And the hardware seems to be fine - the render is lightning fast (renders the startup scene in under .1 second; it's just laggy when loading the uv screen), so I don't know why it's like this, or what to do. Also, I ran Blender 2.77 on this computer, and it was also slow (although it was fine on my previous computer).  It could be an issue with drivers perhaps (or maybe OpenGL?), but I ran Unreal Engine 4, and had none of the same issues - the speed was pretty decent on that; so I'm not sure why Blender would be so slow. Any help is appreciated - thanks guys! Thought Experiment - Poking a stick across a Black Hole's Event Horizon The classical explanation of a black hole says that if you get to close, you reach a point - the event horizon radius - from which you cannot escape even travelling at the speed of light. Then they normally talk about spaghetti.  But here's a thought experiment. What if I have a BH with event horizon radius R such that the gravitational gradient at the event horizon is far too weak for creating pasta. I build a Ring with radius R+x around the BH. Then I lower a pole of length x+d from my ring towards the BH, so that the tip passes beyond the event horizon.  Now what happens when I try to pull the pole back? Is there a single word for something that is simultaneously not interesting and not boring? I am looking for a word that would describe a concept or activity, such as playing a game X, that is1 neither boring nor interesting. In other words, it should mean ‘neutral’ wrt intrestingness (~ boringness) but still comparable on the scale of intrestingness to both ‘boring’ and ‘interesting’.  An example sentence would be  As far as I'm concerned, long-distance running is [insert word]; it is not boring but it's also not interesting.  Criteria2   Has to be unambiguous3, and may consist of two words at most. Has to be a single word. Should be formal in the sense that it is included in Cambridge Dictionary, or Merriam Webster, or both. Should not contain negating prefixes, such as ‘un-’.   Does such a word exist in the English language?  For readers' convenience, please state which criteria you believe your word suggestion has met. You may do so, e.g., by adding a disclaimer at the top or bottom of your answer. Simply writing (1., 2.) is sufficient.        1 Assume a ternary system of intrestingness.   2 If not all criteria can be met, you may relax  the requirements from bottom to top, in the following order: 4. -> 3.  -> 2. -> 1. Answers with the smallest number of relaxed criteria in this order will be preferred in accepting. For example, fulfilling  (1., 2., 3.) is better than achieving (1., 3., 4.). Situations where a  higher priority ordering fulfills fewer criteria than a lower priority  vector, such as (1.) vs (3.,  4.), will be assessed on a case-by-case  basis.   3 Words such as ‘unappealing’ thus even fail  criterium 1 because in a three-valued system ‘unappealing’ ~ ‘not  appealing’ ~ ‘not interesting’ can be taken to mean either ‘boring’ or  ‘neutral wrt intrestingness’. What are some practical examples of a constant current source? Please don't state the obvious that nothing is a perfect constant current source.  And please do not give examples of how to create them from components.  I looked at this  which stated the obvious - that you can not have a perfect current source.  So when I say constant current source, I am not referring to the mathematical concept used in circuits, but "natural" components that provide a close approximation to this. Key word is approximation.  A bit of research pulled up one interesting example which was a solar cell.  Is this a valid example?  A google search pulls up a similar Quora question . Which gave the example of a solar cell.  This is a more technical site and I was looking for a more engineering answer not a mathematical or laymen's answer.  Thanks. Destroying a Planewalker before its ability is used Say my opponent plays a planeswalker during their first main phase.  I have a  or some other effect that will kill it after it is already in play, but before they use any of its abilities.  I could respond to them using a +1 ability, but I want to kill it before that effect goes on the stack.  In most casual situations, it would basically come down to whether I say "Hero's Downfall" first, or they say "+1" first.  But I can't imagine that's how it's supposed to work.  Is there a time between when the planeswalker hits the battlefield and when they activate its abilities that they must let me respond? It seems to me that situations like this arise frequently, and they are resolved simply by who speaks up first, which is horrible for such a rules-oriented game. I have just installed 12.04 LTS on my laptop and I'm kinda new in this. I am required to write some program in Fortran 77 and I need a Fortran 77 compiler. I have tried installing g77 as described in the Ubuntu website but was unable to do so. I have typed in the following in terminal:  sudo apt-get install g77   and get the following:  Reading package lists... Done Building dependency tree        Reading state information... Done Package g77 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'g77' has no installation candidate   May I know what is the problem? I have tried searching g77 in the software center but apparently it is not there. Does this mean it is not available for 12.04?  I'm currently using an Asus K43S laptop with Intel Core i5-2450M processor and my Ubuntu is installed alongside Windows. Add table row in jQuery What is the best method in jQuery to add an additional row to a table as the last row?  Is this acceptable?  $('#myTable').append('&lt;tr&gt;&lt;td&gt;my data&lt;/td&gt;&lt;td&gt;more data&lt;/td&gt;&lt;/tr&gt;');   Are there limitations to what you can add to a table like this (such as inputs, selects, number of rows)? I receive the error 'grub-install /dev/sda failed' while attempting to install Ubuntu as the computer's only OS. I am attempting to install Ubuntu on a box which was previously running Windows 7.  I have also experienced the dreaded "Unable to install GRUB" error.  I am not attempting to dual boot. I have previously run a Windows boot disk and removed all existing partitions.  If I run the Ubuntu 12.04 install CD and click install after the config screens, I get the error   Executing 'grub-install /dev/sda' failed. This is a fatal error. (It is the same error as this question: )  All the questions I've read while looking for a solution are related to dual boot.  I'm not interested in dual boot, I'm after a clean out the box Ubuntu install. How can I achieve this?  (For my sanity, please use very simple instructions when responding. I don't claim to have any talent either for linux or as a sysadmin)    Additional details copied from comments dated: 2012-05-29 ~15:19Z  After booting from the CD, clicking Try Ubuntu, and then sudo fdisk /dev/sda I get fdisk: unable to seek on /dev/sda: Invalid argument  sudo fdisk /dev/sdb gives   Device contains neither a valid DOS partiion table, nor Sun, SGI or OSF disklabel.  Building a new DOS disklabel with disk identifier 0x15228d1d.  Changes will remain in memory only until you decide to write them.  After that of course, the previous content won't be recoverable.  Warning: invalid flag 0x0000 of partition table 4 will be corrected by w(rite).  Command (m for help):   I should add the Live CD desktop is graphically bad. I've got missing parts of programs and the terminal occasionally reflects to the bottom of the screen. But I can't imagine this is related. How to connect to my web server? What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? There are many spells which explicitly target creatures:     Choose one creature within range   The target creature is normally supposed to make a saving throw in this case.  What happens when the caster chooses a creature, that is actually an object? For instance, a scarecrow, a mannequin, an illusion, or a corpse.  Examples:   An evil mage is hiding in the bushes. He casts an illusion of himself on the road, using the Silent Image spell. A PC sorcerer walks down the road, sees the illusion and casts Magic Missile on it. A cleric is standing near a pile of dead bodies. He has suspicions that a few corpses are actually undead creatures. He casts Sacred Flame on every corpse, see which if them succeed, hence, reveals the undead. A wizard casts Acid Splash on a mannequin, trying to damage it with acid.   Possible outcomes I can think of are:   The caster cannot even try to cast the spell, regardless of their information about what the target actually is. The caster tries to cast the spell, the casting fails, not expending the spell slot. The caster casts the spell, it expends the spell slot, then fails. The caster casts the spell, but it hits the original and not the illusion. The caster casts the spell, it flies to the target illusion but deals no damage to it (passes through it).   References in the rules and Sage advice  I was trying to investigate, but it didn't make things clearer:    has an answer, that Magic Missile won't hit. It says nothing about if it can be targeted at all, it leaves it up to the DM. Magic Missile automatically finds the original, even when the caster targets an illusion. At least . Now . This supports the counter-intuitive option 4.   on the other hand   , if there were no perceivable effects (let's say you can try to cast the Command spell on an illusion). That assumes that casters can waste their spells on illusion, which means they at least can target it. This supports options 3 and 5. Why do RAW images in Darktable have a lot of noise? Hide the term id on autocomplete widget Most efficient way to check for DBNull and then assign to a variable? This question comes up occasionally, but I haven't seen a satisfactory answer.  A typical pattern is (row is a DataRow):   if (row["value"] != DBNull.Value)  {       someObject.Member = row["value"];  }   My first question is which is more efficient (I've flipped the condition):    row["value"] == DBNull.Value; // Or   row["value"] is DBNull; // Or   row["value"].GetType() == typeof(DBNull) // Or... any suggestions?    indicates that .GetType() should be faster, but maybe the compiler knows a few tricks I don't?  Second question, is it worth caching the value of row["value"] or does the compiler optimize the indexer away anyway?  For example:    object valueHolder;   if (DBNull.Value == (valueHolder = row["value"])) {}   Notes:   row["value"] exists. I don't know the column index of the column (hence the column name lookup). I'm asking specifically about checking for DBNull and then assignment (not about premature optimization, etc.).   I benchmarked a few scenarios (time in seconds, 10,000,000 trials):  row["value"] == DBNull.Value: 00:00:01.5478995 row["value"] is DBNull: 00:00:01.6306578 row["value"].GetType() == typeof(DBNull): 00:00:02.0138757   Object.ReferenceEquals has the same performance as "=="  The most interesting result? If you mismatch the name of the column by case (for example, "Value" instead of "value", it takes roughly ten times longer (for a string):  row["Value"] == DBNull.Value: 00:00:12.2792374   The moral of the story seems to be that if you can't look up a column by its index, then ensure that the column name you feed to the indexer matches the DataColumn's name exactly.  Caching the value also appears to be nearly twice as fast:  No Caching: 00:00:03.0996622 With Caching: 00:00:01.5659920   So the most efficient method seems to be:   object temp;  string variable;  if (DBNull.Value != (temp = row["value"]))  {       variable = temp.ToString();  }