Let $(a_n)$ be a sequence in $\mathbb{R}$ and let $b_n = a_{n+1} - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.
I have a question that says "Show that there is a bounded sequence $x_n$ which is not convergent but has the property that $x_n - x_{n+1} \to 0$ as $n \to 0$.  What does this mean? Do I need to come up with an example or does the problem actually want me to prove such proposition?  By the way, I see that this sequence looks like Cauchy because of $x_n - x_{n+1} \to 0$ as $n \to 0$, but it is obviously not.
Pointer vs. Reference What would be better practice when giving a function the original variable to work with:  unsigned long x = 4;  void func1(unsigned long&amp; val) {      val = 5;             } func1(x);   or:  void func2(unsigned long* val) {      *val = 5; } func2(&amp;x);   IOW: Is there any reason to pick one over another? Examples of topologies in which all open sets are regular? An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia . Furthermore, the regular open subsets of a space (any space) form a complete Boolean algebra.  I'm coming to this from logic and algebra, with not much background in topology. I can't figure out which topologies have "interesting" collections of regular open sets. For example, in the trivial topology and the discrete topology, every open set is regular, if I'm not mistaken. Those are not "interesting" topologies. I assume there are other topologies in which the space X and the null set are the only regular open sets. If so, those aren't "interesting" either, at least not with regard to their regular open ets.  I believe what I'm looking for is topologies in which every open set is regular, other than the ones I've just described.  Thanks for any help. I messed up my system earlier, I was greeted with a black screen, when booting in to Ubuntu. When I started up my laptop, I selected the recovery option from the grub menu, and chose fallback at root terminal. I saw that I was able to use the add user command, with it, I probably could use to create a privileged user on my machine.   Isn't that a security issue?  One could have stolen my laptop and at startup chose recovery and add another user, I'm fudged then. Including my data.  Come to think of it, even if you somehow remove that entry, one could boot from a live-CD, get a chroot up and running and then add another user, with the right privileges that allows it to see all my data.   If I set the BIOS to boot at my HD only, no USB, CD/DVD, Network startup, and set a BIOS password, it still wouldn't matter, because you'd still have that grub recovery startup entry.  I am fairly certain that someone from China, Russia can't hack my Ubuntu Trusty Tahr, from the network, because it's secure like that. But, if one has physical access to my - your - machine, then, well, that's why I'm asking this question. How can I secure my machine so that hacking through physical access is not possible?    Bug Report: What is a word for deliberate dismissal of some facts? Java. Splitting a multiple word string into two word strings every space Partition on sublists beginning with a certain marker elements PostgreSQL create line from points Using apt-get how to check have installed latest version ? Hello im just new for Linux   I know that  sudo apt-get   is for installing application   but how I would know that I have installed the latest version of the application   for example apt-get install sun-java6-jdk   is there an option to upgrade or see the latest version   or its ok to install an old version the application and Linux will upgrade it for me ? Get TEXT value of a CLOB OID in Postgresql Adjust space between entries in LoF and LoT Coauthoring with a reviewer possible? I recently submitted an article to a math journal and got a response from a reviewer, with an attachment. In it, I have found some good  advice that could improve my article tremendously.  I am a bit suspicious of its accuracy since it could resolve what I have been working with for some months in a few  sentences.   But anyway, it was enough for me to ask the reviewer for coauthoring with me.   My question would be, does it often happen that an author asks a referee for coauthoring during the review process? Or could it even be possible?  I was given a month to revise my article, but the chance that I could revise successfully by myself seems to be fat.  I would be looking forward to any advice.  Thanks in advance. "Images with street signs" CAPTCHA impossible to solve I just got challenged (over on ) by the new "Mark all images with street signs" CAPTCHA.  I must have tried a dozen times to solve it by checking all images that had street signs, but I never succeeded.  Of note is a text input field without instructions of what to enter there. I suspect that is the problem: It is expecting input, but doesn't tell me what to input.  It's either broken or I am missing something.  I am using Chrome on an iPhone in "full site" mode. Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? I've read the faq's and they all seem to be the same to me - Just Q &amp; A on a variety of topics.  What's the difference in a nutshell? If I have a software engineering question, which site is best for me to use? Java: how to parse double from regex From what I understand, entropy is a concept defined by the experimentalist due to his ignorance of the exact microstate of a system. To say the number of accessible microstates $W$ of the universe is constantly increasing is nothing more than saying 'ignorance begets ignorance'.  I have often encountered the argument of ever increasing entropy for the presence of an inherent time asymmetry, most prominently in the works of Penrose. It just doesn't seem to make sense.  Let us imagine alien beings who are experiencing time in reverse. For them, increasing ignorance is in the direction of our decreasing ignorance. So how can perpetual increase in entropy indicate an 'arrow of time'?  One possible explanation (I thought of) for this argument was the fact that there may never be a mechanism to reduce ignorance. I will pose this as a question:  An observer determines the number of possible microstates of system+observer to be $W_0$ at time $t_0$. After improving his measurements, can he (at a later time) measure $W'$ (where $W_0&gt;W'$) as the number of possible microstates? Assume that all the microstates are equally probable in this case.  Unless the above in false in general, how else can anyone claim that entropy reveals an arrow of time?  EDIT: I am performing this discussion for isolated systems (the universe or it's parts if relevant). The means I propose to reduce entropy in the present, is by redefinition of macroscopic variables and of microscopic models used to count the number of distinguishable micro-states (that obviously yield the same macro-state). This is essentially an argument against the robustness of entropy via redefinition. Where do I get the key for the first house? I've 'completed' the game, but didn't find the key for the first house. (I got the one for the arcade machine house). Where can I find it? population and subsamples : mean equality? I want to perform a two sample test, to see whether theirs means are equals. One of the sample is rather big (#60000 values) and represents the scores of a whole population. This population is partitioned in small classes of less that 1000 people. The other sample of the test is a subsamble of the former. It corresponds to people of the same class. So, each time, there is less than 1000 values in it. I have no idea about the variances, so I'd think of using Welch's t-test.  But I'm concerned regarding the independence hypothesis: can it be valid, considering that the second sample is negligible when compared to the overall population? Taking Seats on a Plane Understanding Euclid's proof that the number of primes is infinite. In Euclid's proof, if $p_1, p_2, \dots, p_n$ are the only primes then $p_1 \times p_2 \times \dots \times p_n + 1$ is not divisible by any of $p_1, p_2, \dots, p_n$ (because of some algebraic facts), which makes another prime and is a contradiction.   The proof makes sense logically, and I tried some numerical examples to "feel" the proof better but...  $2 \times 3 \times 5\times 7\times 11\times 13+1$ is not a prime! $2 \times 3 \times 5\times 7\times 11\times 13 \times 17+1$ is also not prime! Why is the general case proof is not working for these examples? Executing a certain action for all elements in an Enumerable