$f:[0,\infty)\rightarrow\mathbb{R}$ is differentiable and $\lim_{x\to\infty}f'(x)=0$, prove $\lim_{x \rightarrow \infty}[f(x+1)-f(x)]=0$.
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.
Where is a good place to learn classical optics for high school competitions? I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue. What Is Exponentiation? Going through a text file line by line in C When doing an apt-get update/upgrade I see that warning  W: Conflicting distribution: https://downloads.plex.tv/repo/deb public InRelease (expected public but got )   I was upgrading to Ubuntu Zesty 17.04 before and do not know how to fix the above. Already have seen ? but there are no suggested fixes for that.  Maybe one of you can give some hints? Thanks! (Before anyone jumps to the conclusion that this is the same question as , please read it through. Also I am looking for more comprehensive and detailed data than the poster in .)  Many years ago I used to use a tool called Line Speed Meter (or Monitor) by TCPIQ. This was a brilliant piece of software that would sit quietly in the background and several times a day (according to a selectable frequency) would reach out and test internet line performance against servers all over the world. In the course of a week (or more), thousands of servers would be tested and enough data would accumulate that one could analyze performance according to time of day, day of week, location of server, etc. Drill-down was possible to the point where you could see every server that was tested.  This data was invaluable in determining if one really was getting the performance being paid for. It was also infinitely more valuable than spot checks you perform at sites like  or even the better-regarded ones like  or  (both of which, incidentally, are biased toward USA users - which I am not one of anymore).  Unfortunately Line Speed Meter is no longer supported and the company that made it now has instead a tool called isposure that is nowhere near as flexible or transparent - it is geared to supplying data to ISPs rather than to the consumers. For example, according to their FAQ, it usually runs only once a day automatically - I'd like to start with running it every couple of hours for a week or two - and scale back once I have a good basis so that I can notice when performance changes. Also, the details of the tests are no longer visible to the consumer-user.  My question: is there a (preferably free) tool/service available today to automatically collect comprehensive line speed metrics so I can map out and analyze the performance I am getting on my Broadband internet connection?  EDIT 2012-02-13: I left something out of the above.  In my country, each ISP sells a package that implies performance beyond the borders of my small country. They own (or lease) the lines going across the ocean. They have full control of the bandwidth leading from their servers to the backbones in the USA and EU. Peformance through the cloud is very much relevant and although not entirely, certainly largely under the local ISP's control, since traditionally, the bottleneck has been traffic out of the country.  As an example, at the moment, my ISP is advertising widely their new transatlantic cable and how their performance now outstrips everyone else's. I haven't "felt" it - so I want a tool to quantify it.  Performance within my country is usually good. The problem is that almost all my work involves access to the Internet in North America, Europe, Pacific Rim and others. So I care about and want to document the end-to-end performance I am getting from my ISP - and if I switch, compare to what I get at a new ISP.  EDIT 2013-03-07: As @harrymc pointed out I neglected to mention OS. All the machines on my home LAN are (sadly) Windows (XP or 7). If A tool requires *nix, I have a couple ancient machines in storage I could probably re-purpose - but my preference is for a Windows-based tool. Thanks.  EDIT 2013-03-10: As a further clarification - a method to monitor speeds achieved during regular work provides half of what I'm looking for. What was nice about TCPIQ is that it would ferret out thousands of sites around the world and do brief d/l &amp; u/l tests against them to check speed and then log the results. This gave a valuable statistical database for analysis of speed overall, as well as during times of day, days of week and according to geographical targets. That's the sort of functionality I'd like to achieve (again). What's a big-vocabulary word for someone with a big vocabulary? What's a big-vocabulary word for someone with a big vocabulary?  I'd like to say I'm looking for a "_____". Lightning component scripts loading issues since Spring 17 update Showing that a sequence is monotone and thus convergent Here's the original question:     Let $(a_n)$ be bounded. Assume that $a_{n+1} \ge a_{n} - 2^{-n}$. Show   that $(a_n)$ is convergent.   Okay, I know that if I can show that if the sequence is monotone, I can conclude that it is convergent. But I am not sure how to show that it is monotone.  I know that  $$a_n \le a_{n+1} + \frac{1}{2^n} &lt; a_{n+1} + \frac{1}{n}$$  It looks to me as if it is monotonically increasing but I'm quite not sure how to prove my claim. Any hints would be appreciated. Find the subgroups of A4 I had a question I was hoping for some help on:     Find all of the subgroups of $A_4$   Here is what I know:  $A_4$ is the alternating group on 4 letters. That is it is the set of all even permutations. The elements are:  $(1), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143), (234), (243)$   which totals to 12 elements. Which means, the subgroups should have order 1,2,3,4,6 and 12. The groups of order 1 and order 12 are trivial. I also know the groups of order 2 which are:  $[1,(12)(34)], [1, (13)(24)], [1, (14)(23)]$  but would someone be able to help me find the subgroups of order 3, 4 and 6 and be able to provide a good, easy explanation of how you got to them for me? I'm pretty sure there is no such groups of order 6, but I don't know how to explain why.  Thank you so much for your help in advance, I really appreciate it! Changing an element's ID with jQuery I need to change an element's ID using jQuery.    Apparently these don't work:  jQuery(this).prev("li").attr("id")="newid" jQuery(this).prev("li")="newid"   I found out that I can make it happen with the following code:  jQuery(this).prev("li")show(function() {     this.id="newid"; });   But that doesn't seem right to me. There must be a better way, no? Also, in case there isn't, what other method can I use instead of show/hide or other effects? Obviously I don't want to show/hide or affect the element every time, just to change its ID.  (Yep, I'm a jQuery newbie.)  Edit I can't use classes in this case, I must use IDs. Adding values to a C# array Probably a really simple one this - I'm starting out with C# and need to add values to an array, for example:  int[] terms;  for(int runs = 0; runs &lt; 400; runs++) {     terms[] = runs; }   For those who have used PHP, here's what I'm trying to do in C#:  $arr = array(); for ($i = 0; $i &lt; 10; $i++) {     $arr[] = $i; } Best way to clone an installation (copying to identical hardware) Kind-of like  but slightly different (I think), in that I have 6 identical Acer Aspire Revo R3610 machines. One is (almost) configured to my requirements - when I'm done preparing it I'd like to make the other 5 machines absolutely the same. I'm very new to Ubuntu, what's the most straightforward (easiest) way of doing this?  The machines are going to live on different networks if that might otherwise be a problem (eg with Windows you can clone disks but you then have to make registry changes afterwards if they're going to run on the same network etc). The hardware in all 6 machines is, I stress, the same!  How can I efficiently clone one source image on to these identical machines?  Please restrict one software/solution per answer Is it possible to backup individual files with Deja Dup? In the Deja Dup preferences, it is possible to select certain folders to backup and certain folders to exclude.  I would like to be able to backup individual files in my Home folder (Such as .vimrc and .bashrc), but I don't see any way to do this.   So my question is, is it possible to either     Select individual files (instead of whole folders) to backup, or backup my Home folder, but exclude all subfolders? How can I selectively disable paste blockers Too much whitespace in "Thanks for the feedback" While attempting to downvote something on a site I don't have an account on, I see this message, which has too much whitespace:    The same thing happens with upvotes (and プラス票が):    (I think it's safe to say that this is a network-wide bug, since I took these all on different sites.)  However, it looks more reasonable when you are not logged into the network at all (e.g. in private browsing):    An earlier version of this issue was reported at , but the difference is that there is no newsletter UI on the right anymore. (The extra space could have been justified before given the height of the newsletter; now it's definitely a bug.)  Also, is newsletter supposed to be gone? How to hook into the CPT's title placeholder? MacBook Pro crashes randomly after dropping it on the floor A few days ago my MacBook Pro slipped out of my hands and fell onto the tile floor. The outside got banged up quite a bit and the Touch Bar no longer lights up but still responds to touches sometimes but not all the time.   Anyway, since then it will randomly restart every once in a while with a kernel panic. I have the log posted below and wanted to know if anyone could tell me what is causing the panic? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? Representation of electromagnetic wave? In Representation of electromagnetic waves we show both electric and magnetic fields perpendicular to each other and having zero magnitude at one point and maximum at other point. This representation and understanding of EM waves violating law of conservation of energy... I'm wondering what the preferred method is for dealing with AJAX calls. Should one use the same plugin php file to process the POST or a separate one? Which is cleaner or safer?