I have that convergence requires that the following be true   $$ \forall\epsilon&gt;0\exists N: n\ge N:|a_n-a|&lt;\epsilon $$  for some sequence $\{a_n\}_{n\in\mathbb{N}}$ that converges to $a$ for $n\to\infty$.  I want to use the reverse of this requirement. I do not know how to express it but I am guessing  $$ \exists\epsilon&gt;0\forall N:n\ge N:|a_n-a|\ge\epsilon $$  it is expressed like the above. Can you help me out?
My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$"  So, I have my definition of a convergent sequence: "$\forall\varepsilon&gt;0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|&lt;\varepsilon$ $\forall n \in \Bbb N$ with $n&gt;N$"  Would the correct negation of this be "$\forall\varepsilon&gt;0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|&gt;\varepsilon$ $\forall n \in \Bbb N$ with $n&gt;N$"?  It doesn't seem that this is the answer as the next part of my task is to prove that a sequence is divergent using my formed proof, but it'd be difficult to do since it's a general proof of divergence and not just a proof that $(a_n)$ doesn't converge a specific number $\mathscr l$  Perhaps I should find a prove that $(a_n)$ tends to $\pm\infty$? This is more simple but it does not include monotone sequences such as $x_n:=(-1)^n$.  Can someone assist me with this task? All comments and answers are appreciated.
Compare double in VBA precision problem I have trouble comparing 2 double in Excel VBA  suppose that I have the following code  Dim a as double Dim b as double a = 0.15 b = 0.01   After a few manipulations on b, b is now equal to 0.6  however the imprecision related to the double data type gives me headache because  if a = b then  //this will never trigger end if   Do you know how I can remove the trailing imprecision on the double type? Can I permanently block an update (non-driver) that will not install on my machine on Win10 pro? I have an update to Visual Studio 2010 that will NOT update and has failed hundreds of times. Manually installing it also fails, and no source has been found for manually getting this very old patch to install on Windows 10. I believe the problem is that Microsoft simply hasn't re-tested KB2890573 at all and it doesn't install on Win10 ver 1511 at all, anymore.  I do not want to use the MS utility that will postpone this update for a while. I want to permanently block it so it does not appear in the windows update list ever again. Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$.   I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any case, any sort of help is appreciated. Custom function won't refresh as inputs are changed Why is this function continuous, unlike the Dirichlet function? $f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$ I get Chrome notifications from several websites and they work when I don't even have any tabs open for those sites.  But sometimes I get these Chrome notifications when I am not at my desk or looking at my laptop.  As some of these notifications are useful to me, I would like to be able to access any that I have missed.  Is there a way to view the notification history? Which side of wave-particle duality to choose in a given situation Operator precedence in a find command? See below example:  $ ls -1p appledir/ applefile orangedir/ orangefile $ find . -type f -name apple\* ./applefile $ find . -type f -name orange\* ./orangefile $ find . -type f -name apple\* -o -name orange\* ./applefile ./orangedir ./orangefile $ find . -type f \( -name apple\* -o -name orange\* \) ./applefile ./orangefile $    I was surprised to discover I needed the parentheses for this to work as expected; apparently I haven't internalized the rule of precedence by which find evaluates its arguments.  How can I easily predict when I will and when I will not need to use parentheses to explicitly group find primaries?  Put another way, what are the rules by which I can imagine find inserting parentheses into the commands I give it, which will allow me to accurately predict how it will evaluate ungrouped expressions? How do you transition from 1-page dungeon map to battle mat? I have a bunch of different ways I use to make maps for my tabletop game.  Sometimes I draw one on graph paper.  Sometimes I use .  Sometimes I copy a map from a published adventure or use the  random generators.  These all have the same output:  a piece of paper (or maybe two) with a map on it, and the map is usually quarter inch squares, or so.  When it's time to put minis on a mat, though, I need to copy over the relevant details of the map to our battle mat.  (We use Tact-Tiles.)  This is fine when most of our map is overland detail and I'm just copying down one or two rooms in which combat will occur.  For dungeon crawls, where there are more random encounters or where combat can chase through many rooms, it can lead to a pretty tedious drawing of the whole dungeon as we go.  I feel foolish for having to ask, but is there some trick, technique, or something to make it less tedious to transcribe maps from printed pages to the battle mat?  EDIT:  Answers in general are great, but I should note that the best answers, for me, will be informed by this previously-unmentioned fact:  the game venue is not my house, so I have limited prep time on site. Where are some good places to find royalty free stock images? iPhone stuck on Connect to iTunes My iPhone is stuck on Connect to iTunes Screen (support.apple.com/iphone/restore iTunes logo and up arrow with lightning cable) I have Tried :-   Connecting it to iTunes and Updating, It updates Successfully.But on the same screen again   Restoring, It restores successfully.But on the same screen again  Tried Recovery Mode Screen Method and DFU method Tried Closing and Restarting iTunes Tried Deleting 'iPhone Software Update' folder in iTunes and Re-Downloding Updates    Nothing Works... Any Help will be appreciated What's your recommendation on drive partitioning schemes for a desktop and home server? What partitioning scheme do you recommend for a desktop? I've always created three or four primary partitions -- root, swap, home, and sometimes a separate boot partition. Ubuntu's default install offers LVMs. I've never had to add additional drives or space, so it never seemed like a big deal. Whenever I do a fresh install, though, I always think there might be a better way. Natural (human alpha-numeric) sort in Microsoft SQL 2005 We have a large database on which we have DB side pagination. This is quick, returning a page of 50 rows from millions of records in a small fraction of a second.  Users can define their own sort, basically choosing what column to sort by.  Columns are dynamic - some have numeric values, some dates and some text.  While most sort as expected text sorts in a dumb way. Well, I say dumb, it makes sense to computers, but frustrates users.  For instance, sorting by a string record id gives something like:  rec1 rec10 rec14 rec2 rec20 rec3 rec4   ...and so on.  I want this to take account of the number, so:  rec1 rec2 rec3 rec4 rec10 rec14 rec20   I can't control the input (otherwise I'd just format in leading 000s) and I can't rely on a single format - some are things like "{alpha code}-{dept code}-{rec id}".  I know a few ways to do this in C#, but can't pull down all the records to sort them, as that would be to slow.  Does anyone know a way to quickly apply a natural sort in Sql server?    We're using:  ROW_NUMBER() over (order by {field name} asc)   And then we're paging by that.  We can add triggers, although we wouldn't. All their input is parametrised and the like, but I can't change the format - if they put in "rec2" and "rec10" they expect them to be returned just like that, and in natural order.    We have valid user input that follows different formats for different clients.  One might go rec1, rec2, rec3, ... rec100, rec101  While another might go: grp1rec1, grp1rec2, ... grp20rec300, grp20rec301  When I say we can't control the input I mean that we can't force users to change these standards - they have a value like grp1rec1 and I can't reformat it as grp01rec001, as that would be changing something used for lookups and linking to external systems.  These formats vary a lot, but are often mixtures of letters and numbers.  Sorting these in C# is easy - just break it up into { "grp", 20, "rec", 301 } and then compare sequence values in turn.  However there may be millions of records and the data is paged, I need the sort to be done on the SQL server.  SQL server sorts by value, not comparison - in C# I can split the values out to compare, but in SQL I need some logic that (very quickly) gets a single value that consistently sorts.  @moebius - your answer might work, but it does feel like an ugly compromise to add a sort-key for all these text values. Setting background-image using jQuery CSS property Detecting start bit in software UART I'm experimenting with writing a software UART on my microcontroller using GPIO pins. This is to temporarily add a UART channel on a project until we get the new design implemented that uses a uC with more UART ports.  What I'm having difficulty with is correctly detecting a start bit in a serial stream. The source of the stream is external and doesn't care when my device powers up. So it's very likely that my device will power on and start seeing data bits in the middle of a byte transmission. Undoubtedly, that will cause my software UART to read erroneous values, as it won't be able to tell the difference between a start bit and any other high-to-low transition.  Is this an inevitable issue with a UART channel? Or is there some clever trick that the uC manufacturers use in their hardware UARTs? Antonym of "deepen" Since there is no word "shallowen", is there a good antonym of "deepen"? The meaning of "deepen" that I am referring to here is "to cause to become deeper." It doesn't necessarily refer to digging a hole. For example, a temporary deformation in a soft surface:     Putting the anvil on the mattress deepens the depression in it, but removing the anvil ______ it.   I would also be happy with an antonym of the intransitive version "to become deeper." I need a list of integers from 1 to x where x is set by the user.  I could build it with a for loop eg assuming x is an integer set previously:  List&lt;int&gt; iList = new List&lt;int&gt;(); for (int i = 1; i &lt;= x; i++) {     iList.Add(i); }   This seems dumb, surely there's a more elegant way to do this, something like the Since my previous questions were not answered I will try to ask again more precisely.  I previously had the Nvidia driver installed and working correctly.  Then the ubuntu software update did something which broke it.  In fact, it completely de-configured it so that the standard display driver is used when the computer reboots.  I would like to re-enable the existing Nvidia  driver without re-installing the Nvidia driver.  How do I do that?  This machine is no longer connected to the internet.  NOTE: I am not asking how to disable automatic updates in this question.  I asked that elsewhere (and it wasn't answered).  NOTE: I am not asking how to stop the Ubuntu software updater from breaking the Nvidia driver the next time it updates the system.  That would be nice to know, but I asked that already and did not receive the correct answer either.  NOTE: Please don't make this as duplicate simply because someone else asked a question about Ubuntu and the software updater.  Try to read the actual question and see how it's different and try to realize how annoying it is when users have to spend hours of their own time to correct the problems that other people causing by incorrectly designing software updaters. Logic Behind StackOverflow Question IDs