I am not even sure where to start with this one. I understand for something to be a limit then that means if we have a sequence $\{a_n\}$ of irrational numbers then for any number $a$ in the reals, the $|a_n - a| &lt; \epsilon$ for $\epsilon &gt; 0$. But how would you show that any number fulfils this statement?
Show that every irrational number in $\mathbb{R}$ is the limit of a   sequence of rational numbers. Every rational number in $\mathbb{R}$ is   the limit of a sequence of irrational numbers.   How can I prove this?
How to navigate to C drive in bash on WSL-Ubuntu? I'm using retextured snowballs and eggs for the different paintballs for two teams in a paintball map, so I can't replace the eggs with snowballs.  I have tried using /gamerule doMobSpawning false, but that still allows the chickens to spawn. I have also tried using /tp @e[type=Chicken] 0 0 0 on a repeating command block to quietly remove the chicken, but there's still a very noticeable frame where the chicken is visible. /effect @e[type=Chicken] minecraft:invisibility 10 1 false isn't fast enough, either.  I'm not using any plugins or mods.  tl;dr Without snowballs, how can I stop chickens from appearing for even a single frame. What's the fastest way to make money online? Everything in the online world is expensive, well the best things are. We all want the luxurious apartment with 10 cars garage and the best weapons but to achieve that, we need money. Which missions or activities give the most cash ? How can I make all text the same font size (normal text, footnotes, acknowledgements etc.)? I have to submit a manuscript to a journal, and I just realized that they want every single part of the paper to be in 12 pt font. This includes tables text, footnotes, acknowledgements etc. Is there a way to force all text to be the same size? Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$? In my textbook on elementary number theory from a class last year, as well as elsewhere through my academic experience and even posts here, I often see the greatest common divisor notated as $(a,b)$ (where this represents the greatest common divisor of $a,b$).  My question is, why is such an unusual notation used? I say it's unusual (if not a bit absurd), for a couple of reasons:   It is unintuitive. It does not grant the reader an immediate knowledge of what it means. In particular, I can easily see it being misinterpreted as an ordered pair of points, and not as a function (which it basically is and I touch on in a second) or representing a single value. Good notation would be immediately understood, no? One should not struggle with parsing through the context or meaning. A clearer notation is often used, and just three letters away. You could certainly think of $(a,b)$ as an ordered pair, if we consider it a function. I often see the alternate $gcd(a,b)$: a function, which takes in two inputs ($a$ and $b$), and outputs their greatest common divisor (or factor, if you prefer). Merely adding the three letters makes the notation clearer by clarifying it is  a function, and using a relatively well-accepted acronym. (Or, at least, I often hear it abbreviated "g.c.d." in school.)   I've mulled this over for a while and I can't think of much without borrowing inspiration from other "unusual" notations. For example:   Common understanding in context: consider the $\log(x)$ function. Depending on context, without a base stated as seen here, it can be presumed to mean whatever is most "appropriate." It could be base $e$ (common in mathematics), base $10$ (common in engineering), or base $2$ (common in computer science). The ambiguity is essentially resolved by context in other words - personally, not a fan of that (I like the explicit stating of bases for full clarity), but if such a motivation holds elsewhere, it might stand to reason that $(a,b)$ could just be taken as a shorthand and "understood" in the context of number theory to simply be the greatest common divisor. That it is in fact used despite $gcd(a,b)$ being clearer and somewhat common also does seem to suggest this playing a role. Perhaps it just started that way and it kept being used. For example,  (based on a ) regarding how logarithms, roots, and exponentiation can be unintuitive from a notation perspective, and proposes an alternative, easier to understand notation. I haven't been in middle school in nearly a decade, but guessing from discussions online, that obviously hasn't really picked up too much. In that sense, perhaps it's like that for $(a,b)$: like we still notate exponentiation, etc., the same way today despite how counterintuitive it is (at least for students), perhaps it just "stuck" and proliferated?   I wasn't able to easily find any reason why online, and it bugs me a bit, so I ask:     Why do we so unintuitively notate the greatest common divisor of two numbers $a,b$ as $(a,b)$?   I of course recognize that this is somewhat moot in light of how easily it's made clear - as established above, all I have to do is append the front of the parenthetical by $gcd$ or $gcf$ to make it way better. And of course, people do that. But I'm more curious in where this ambiguous former notation arose and, in light of a better notation, why it is still used today. I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? Using ArcPy to Switch Layout Templates? Using ArcMap 10.1, since there is not a way to control portrait or landscape of the layout view through python, is there a way to switch layout templates without opening a new mxd through python?   Since there is a change layout template button it would be nice to have a python function to do the same. Show that $3p^2=q^2$ implies $3|p$ and $3|q$ IOS 5.1.1 Safemode problem I jailbroke my old iPod Touch 3rd Generation, with IOS 5.1.1, using the latest version of absynthe. It worked for several months, but then it randomly stopped. It went into safe mode, but the touch screen wouldn't work. It showed the lockscreen, but nothing worked at all, except for the physical buttons. The only way to reboot is to follow the same steps as DFU mode (Hold power button 3 sec, then home button AND power for 10, then just home for 8)... However, it always boots right back into Safe mode.  I had to restore, and once rejailbroken, it worked fine, until I installed SBsettings. Then it did the EXACT same thing, and I had to restore and re-jailbreak AGAIN! The other day, I finished installing ALL of my extensive list of tweaks, and got it to about where I wanted it, when lo and behold, it booted to safe mode again. I have spent so much time on this, and I do not want to do it again. Does anybody here have any knowledge on the topic? Is there a program I can run from my computer to manually control the device or kick it out of safe mode? Is there some other method of doing this? I am looking for any solution OTHER than restoring. I have done that several times.  Also, I know the problem isn't with my jailbreak. It worked for several months before, then one day stopped. Now, it has the problem within a day of rejailbreaking, OR instantly if I install SBsettings. I re-installed the jailbreak, and applied the new one as well, and that didn't work. I have given up, and don't want to go through the painful process of restoring, rejailbreaking, and re-installing all of those tweaks again.  Also, because I can't get into my iPod, I can't provide a list of tweaks. I would if I could, but it just can't happen unless I can get in there. Entering and leaving the UK on the day of visa expiry Formation of ammonia from air and moisture On extended bicycle trips I sometimes slept under a plastic sheet, which helped with mosquitoes on hot nights. But there was a noticeable build-up of ammonia which made this almost impossible for more than an hour or so at a time.   Air is mostly nitrogen $(N_2)$ and there is always water (containing hydrogen) on a humid day. But I wonder about the mechanism for the formation of ammonia under these circumstances? There are (according to Wiki) trace amounts of methane and $H_2$ in the air. But methane and hydrogen (g) are I think pretty stable.   I wonder if the presence of exhaled air (containing $CO_2)$ could be a factor?  Thanks for any ideas. SSH login with clear text password as a parameter? I need to login to a user that I've created on a remote host running Ubuntu. I can't use an ssh key because the ssh login will happen from a bash script ran within a server that I won't have access to (think continuous integration server like Bamboo).   I understand this isn't an ideal practice, but I want to either set the remote host to not ask for the password or be able to login with something like ssh --passsword foobar user@host, kind of like MySQL allows you to do for logins.   I'm not finding this in man ssh and  I'm open to any alternatives to getting around this issue. How can I ask a person in which order in his family among the siblings? My colleague he have two brothers. I want to know, he is first/second/third son of his parents, ie., the order in the family among their siblings.  I can simply ask     Are you first son of your family?   It simply give some predictions, I don't want to that.  Is there a better way to ask the question? Integer reciprocals in arithmetic progression Let $m_1,m_2,\ldots,m_k$ be $k$ positive integers such that their reciprocals are in AP. Show that $k&lt;m_1+2$. Also find such a sequence.     Whatever way I tried, whichever formula I used, I could not eliminate the $m_k$ term, which I reckon would be the ideal scenario to find the solution. I am exhausted of any further ideas, please help. Thank you. How to fix the Brazilian daylight saving time (Summer Time) of 2018 in Olson database files? The Brazilian Government has just issued a law that shifts the beginning of the Brazilian DST (Daylight Saving Time) locally called as Summer Time.   This new DST date will come into force in the next month, to be more precise on 18th of November of 2018. So, we won't have enough time to wait for a new Olson update through the operation system updates, neither downloading a new Olson database package might be enough.  I have downloaded the file tzdb-2018e.tar.tz from . Inside that package I found the file southamerica and I have located the following entry:  ... # From Steffen Thorsen (2017-12-18): # According to many media sources, next year's DST start in Brazil will move to # the first Sunday of November, and it will stay like that for the years after. # ... https://www.timeanddate.com/news/time/brazil-delays-dst-2018.html # From Steffen Thorsen (2017-12-20): # http://www.planalto.gov.br/ccivil_03/_ato2015-2018/2017/decreto/D9242.htm Rule    Brazil  2018    max -   Nov Sun&gt;=1  0:00    1:00    - Rule    Brazil  2023    only    -   Feb Sun&gt;=22 0:00    0   - ...   The comments preceding the entry rule for 2018 matches the old law from the end of 2017, but there's a new law that shifts it to the 3rd Sunday of November of 2018.  How to fix that Olson entry shifting it to the 3rd Sunday of November? I have tried Sun&gt;=3 yet, and it didn't work. Are there any non-decreasing securities on the stock market? Are there any securities traded on the stock market (i.e. ETPs) that never decrease in value?    As an aside (in case anyone mentions bonds):  I'm aware there's ETFs for AAA-rated government bonds, but even these dip significantly.  For example, the  has dipped by 9% in a 85-day period before (21/08/2016 to 13/11/2016). Redirect a portion of a bash script to a file to capture set -x output I was checking on something in the Harry Potter wiki and saw Harry listed as "Half-Blood."  Both his parents were magical, but, as we know, his grandparents on his Mother's side were muggles.  This reminds me of the racially loaded terms like octoroon.  (In the south, you still hear those terms every once in a while.)  When there was still segregation, having only 1 great-grandparent of an ethnic group was enough to classify a person of that group.  So how far down the line did it go in the Harry Potter universe?  Just how many generations removed from a muggle-born would a person have to be before they're no longer classified as half-blood or something similar? Fisher's exact test on 2x4 table I am working on 4 different plants. I have the (RNA-Seq) data from sequencing. I look for two events E1 and E2, say, at certain positions in their genome. Let's say E2 is the common one and E1 is a special event. The positions I look for are identical across all 4 (in the genome). And let's say I see that the observation I have for 1 particular position is:      P1   P2  P3  P4 E1   0   20   0   17 E2 100   80 100  120    Here, P1 through P4 refers to plants and E1 and E2 refers to the events. E2 is more common. So, my objective is to actually check if E1 occurs more often in one or more plants than in the others. If they occur at the same proportion in all, of course it is not interesting to me.  I have 2 questions: (I have already asked  before but didn't get an answer)     Will a fisher test be right for this problem?    I hypothesize (Null) that the proportion of E1 is not different in occurrence between the 4 plants. Now, I set out to find if the proportion I have here is by any means significantly different. I use R, fisher.test() and I get p-value=2.5e-10. So, I reject my Null hypothesis for this case because I find strong evidence against it.  Sometimes, I have an observation like this,        P1   P2  P3  P4 E1   0   20   0  17 E2   0   80   0 120    then fisher.test() gives me a p-value of 0.147. So, I don't reject the Null   hypothesis. However, from a biological point of view, I would consider this significant. However fisher test answers the question I originally asked. I guess the proportion 0/0 for P1 and P3 are not useful (or not used).  So, my question is: Is it possible to modify the test such that it is sensitive even if E1 and E2 are 0 in 1 or more plants for a particular observation?     Having thought a bit to frame this post, I guess, in that case I have to ask a different question. How to add a list of vectors (lists) to a directed graph as an edge weight?