The question is as follows:  If $\{a_n\}$ is a convergent sequence of real numbers, then $\lim_{n \to \infty} n(a_{n+1} - a_n) =0 $.  I have been thinking about this question since yesterday but I am unable to prove it. $a_{n+1} - a_n \rightarrow 0$ if $\{a_n\}$ is convergent, but there is also a term of $n$ which is creating trouble for me. I verified this with examples like $a_n = 1/n$ and $a_n =1$ but can't get any idea on how to prove it. Please help me, any idea will be appreciated.
I am trying to prove some theorem, and in the proof of it I have to check the fact that if the sequence $\{a_n\}_{n\in \mathbb{N}}$ converges, then $$\lim_{n\to \infty} n(a_n - a_{n-1}) = 0. $$ I think that this is true because I have tried with many different sequences and it holds, but I do not know how to prove it.  If this is not true, what is a counterexample for it?
$e^A$, where $A= \begin{bmatrix} 0 & \omega \\ -\omega& 0 \end{bmatrix}$ $\omega&gt;0$, Then Compute the matrix:          $e^A$,     where $A= \begin{bmatrix} 0 &amp; \omega \\ -\omega&amp; 0 \end{bmatrix}$      I have never seen any problem like this. So Please help me solving this. Any link for theoretical reading is welcome. Using (or not) the word "to" in a sentence ending with "be able" Which of these is grammatically correct?   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able.   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able to. Adobe Illustrator: Lines with stroke < 2.0pt doesn't change color I created lines using the line segment tool in Adobe Illustrator (latest Adobe Cloud version) and I can't change the color of the stroke (through the Appearance panel) if the stroke weight is less than 2.0pt.   If the stroke is 2pt or more, then the color around the line changes but not the line itself.    The whole behavior is odd for me as I've never faced in Illustrator before.  Edit: added screen below with the line selected. Plot an elliptic curve in Latex Regex for numbers only I haven't used regular expressions at all, so I'm having difficulty troubleshooting. I want the regex to match only when the contained string is all numbers; but with the two examples below it is matching a string that contains all numbers plus an equals sign like "1234=4321". I'm sure there's a way to change this behavior, but as I said, I've never really done much with regular expressions.  string compare = "1234=4321"; Regex regex = new Regex(@"[\d]");  if (regex.IsMatch(compare)) {      //true }  regex = new Regex("[0-9]");  if (regex.IsMatch(compare)) {      //true }   In case it matters, I'm using C# and .NET2.0. mongodb $exists always returning 0 So I understand that Euclidean distance is valid for all of properties for a metric. But why doesn't the square hold the same way? Query Feature and Usage based entitlements license I need to query programmatically the Feature license and usage-based license which is available in UI. Could someone please highlight if we can query the same or is it available in UI only, Thanks. Javascript - alert("zoo">"house") returns true Flex: Figure out when the running swf was compiled? Why do some bad traits evolve, and good ones don't? If a trait would be advantageous to an organism then why hasn't it evolved yet?  Conversely, if a trait is not advantageous or mildly disadvantageous, why does it exist?  In other words why does evolution not make the organism more "perfect"?    This is a general question that would be applicable for any kind of trait. Please keep the answers precise and scientific.   Read this meta post for more information: Javascript: negative lookbehind equivalent? Is there a way to achieve the equivalent of a  in javascript regular expressions?  I need to match a string that does not start with a specific set of characters.  It seems I am unable to find a regex that does this without failing if the matched part is found at the beginning of the string.  Negative lookbehinds seem to be the only answer, but javascript doesn't have one.  EDIT: This is the regex that I would like to work, but it doesn't:  (?&lt;!([abcdefg]))m  So it would match the 'm' in 'jim' or 'm', but not 'jam' I miss those pins I make in Windows.     How can I do the same in Ubuntu 17.10 with GNOME 3? Do I have to worry about copyright issues for code posted on Stack Overflow? I was wondering about the copyright of code snippets from answers. Let's say you ask a question and the solution consists of ten lines of code, which are given in an answer. Are there any restrictions about the use of that code?   It seems that the content of this page is under the . So does one have to attribute the author if code is used in an open source project? Can it be legally used in closed source projects at all?  I guess ten lines of code are not really problematic, but is there a critical number of lines? Automatically Locate Included Images What happens to the view.frame.size attribute after a CGAffineTransform (Rotate)? I want to clear all previous commands from the history of my server. I used history -c and it seems all things are cleared but when I ssh to the server, all the commands are still there.  How can I clear them permanently? Quite simply (and similarly to my other questions about previous versions of this feat  and ) I want to know if this feat is balanced and if not then what should be done to improve it? Since the last iteration I have changed the wording as per David Coffron's answer, increased the time and monetary cost as well as added in the action requirement to activate the rune as per Ruse's answer and increased the costs as per linksassin's answer     Rune Caster      Prerequisites: ability to cast at least one spell of 1st level or higher      When you take this feat you can imbue your spells of 1st level or higher into a rune by casting the spell as a rune, this means that you must have the object you want to put the rune on within reach for the entire duration of the casting as well as any tools required to make the rune.      When casting a rune you add the time listed in the cost time table to the casting time and the cost listed in the cost time table to the components of the spell in accordance with the stored spell.      \begin{array}{|l|l|l|} \hline \text{Spell level} &amp; \text{Time (hrs)} &amp; \text{Cost (GP)}\\ \hline 1 &amp; 1 &amp; 50 \\ \hline 2 &amp; 2 &amp; 100\\ \hline 3 &amp; 4 &amp; 500\\ \hline 4 &amp; 6 &amp; 1,000\\ \hline 5 &amp; 8 &amp; 5,000\\ \hline 6 &amp; 12 &amp; 10,000\\ \hline 7 &amp; 16 &amp; 50,000\\ \hline 8 &amp; 18 &amp; 100,000\\ \hline 9 &amp; 24 &amp; 500,000\\ \hline \end{array}      When making a rune you choose a trigger which is touch and/or command word and can be specified to only a group of people/creatures or a body part. When defining creatures that can activate the rune you may be as specific or general as you want. The DM interprets the designated creatures as your character would have interpreted them when you made the rune.      When you store a spell in a rune some of the ranges are altered, a range of touch becomes a range of self, a range and target keeps the same range and the person who activates the rune designates the target and range with an aoe sets the origin to the center of the rune, lastly a range of self stays the same.      When a rune is activated the spell is released acting as if it had just been cast without requiring any components or concentration. The rune then disappears leaving no trace of its presence. To activate a rune you must complete the trigger, this means speaking any command words and physically touching the rune each if applicable, this requires the activator to use the object interaction action while within reach for touch runes and while within 60 ft for command words. If both triggers are used then they must both be fulfilled within the same turn. What is the best way to determine the number of days in a month with JavaScript? I've been using this function but I'd like to know what's the most efficient and accurate way to get it.  function daysInMonth(iMonth, iYear) {    return 32 - new Date(iYear, iMonth, 32).getDate(); } Fisher Information of a function of a parameter