Consider two real valued sequences $\{x_n\} $ and $\{y_n\} $ satisfying the condition $x_n^3-y_n^3\rightarrow 0$ as $n\rightarrow \infty $. Then as $n\rightarrow \infty$, (A) $x_n - y_n \rightarrow 0$ always (B) $x_n - y_n \rightarrow 0$ only if $\{x_n\}$ converges. (C) $x_n - y_n \rightarrow 0$ only if $\{|x_n|-|y_n|\}$ converges. (D) $x_n - y_n \rightarrow 0$ only if $\{|x_n^2 +x_ny_n+y_n^2|\}$ converges $\bf{Try:}$ Take $x_n=(-1) ^n$ and $y_n=(-1) ^n+\frac 1n$. Then option(B) is discarded. For the other options I don't have clear idea. But I tried to proceed like this- Since $x_n^3-y_n^3\rightarrow 0$ hence for some $n&gt;N$ we have $|x_n^3 - y_n^3|&lt;\epsilon \implies |(x_n-y_n) (x_n^2 +x_ny_n+y_n^2) |&lt;\epsilon$ How to conclude?
Given, $\{x_n\}$ and $\{y_n\}$ are two real valued sequences such that $\lim_{n\to\infty} (x_{n}^{3}-y_{n}^{3})=0 $ Then is it always true that $\lim_{n\to\infty} (x_n-y_n) =0 $ ? I tried by splitting $ (x_{n}^{3}-y_{n}^{3}) $, but that didn't help as $\lim_{n\to \infty} (x_{n}^{2}+x_{n}y_{n}+y_{n}^{2}) $ can be $0$ or non zero or doesn't exist.
How do I create a dynamic key to be added to a JavaScript object variable Please show changed titles separately in edit diffs In an edit diff, when the revision changes the title, removed words are shown struck out in red and added words are shown with a green background. This is fine when there are only removed words, or only added words, but when there are both, the result is difficult to read.        Getting default accessible fields of… No. Getting accessible fields of an object in a Salesforce object… No. Getting accessible fields of a Salesforce object using REST API. Phew. Yeah, ok.        Program causes run. Time error? Oh, Program causes run-time error … when run on specific machine. Yeah, sounds sensible. What was the old title? Oh dear…  Please show both the old title and the new title, one below the other. Lebesgue measurable but not Borel measurable Clerical mistake in UK visa application: should I make a pen-and-ink correction or electronically edit the PDF? I've gone through all similar questions but I believe my specific question doesn't have an answer yet.  I've applied for a Standard Visitor visa. After I completed the application, made an appointment and paid the fee, I noticed that the way I spelled my name on the application differs from how it is spelled in the passport by 1 letter (this is partially because of myself being an idiot, but mainly because my new passport has an incorrect transliteration of my name, whereas my old passport and any other document in my life has the correct one).  So here are the options I see:   Print the application and make a pen-and-ink correction with a black ballpoint pen by crossing the printed name/letter and writing the correct one - as answers to other similar questions suggests. Edit the PDF electronically and print it so that there are no visible corrections. However, as the form is saved at their end, there will be a discrepance between the printed form and saved form with no visible signs of correction.   Which way has better chance for me to have my application accepted?  Also (probably doesn't matter much but still): if it's the pen-and-ink correction, is it better to cross-out the whole name or only the wrong letter? What are best practices/important steps when the Windows Server machine hosting SQL Server 2008 R2 is moved to another domain?   What steps definitely/might need to be done for SQL Server before and after the new domain is joined? How do I log into my Linux laptop if I have forgotten both the username and password? What would happen if every person in the world gathered at a minimum possible area and jumped all at once? Would that have any effect on Earth? Make a momentary switch control a toggle What are the simplest, cheapest, smallest ways to make a momentary switch produce a 2-state toggling output (latching momentary switch)?  In other words, the output is continuously low, and when you momentarily press the button/tact switch, the output changes to continuously high, and then when you press it again, it switches back to low. How to calculate 95% CI for cumulative incidence rate using R? How does Meta Stack Exchange work? Meta Stack Exchange seems to have different rules and user behavior than the "regular" Stack Exchange sites.  How is it different?    Individual questions answered below:                Related Novel(s) based on humanity fleeing from Earth and regressing to a pre-industrial state on a Colony so as to escape detection Shortly before his death, my father spoke to me about a series of novels that he had just finished reading which had a fascinating concept.  However I soon forgot the title(s) and author and I've been absolutely desperate to find out what they were.  The premise (as I understood) was that humanity had expanded out from the solar system into the local star systems before coming into contact with an incredibly aggressive species which was focused on wiping out the entire species.  Earth and her colonies were lost as remnants of the species fled.  Eventually it was decided by the military leadership to set up a colony in order to rebuild the human race.  However, in order for this colony to escape detection and to keep humanity from going back out into space and thus potentially running into this hostile alien race again, a series of satellites were set up around the planet chosen that would lock onto and destroy any form of technology (and usually the surrounding city) that went beyond pre-industrial standards.    One of the series protagonist is a long living android who possesses the consciousness of a human who was present at the original founding of the colony (and who seems shielded from detection) which observes as civilization is repeatedly regrown and torn down again and who attempts to try and reverse humanities self imposed lot in life.      From what I understood this was a series of novels, but I could be wrong. I've been trying to find this for years on my own with only a rough plot line as a guide and it's gotten me nowhere. I have some long URLs which doesn't contain special "line break characters" like "-" or "/". How can I automatically break these URLs?  \documentclass{scrreprt}  \usepackage[ngerman]{babel}                     \usepackage[latin1]{inputenc}                 \usepackage[T1]{fontenc} \usepackage[backend=biber]{biblatex} \usepackage{url} \usepackage{dingbat}  % for \carriagereturn symbol \makeatletter \g@addto@macro{\UrlBreaks}{\UrlOrds} \g@addto@macro{\UrlBreaks}{% \do\/\do\d% } \makeatother  \begin{filecontents}{bachelorarbeit_lit.bib} @ONLINE{java96,     author = {{Joe Black}},     title = {Foobar 1.0},     year = {1996},     month = {1},     url = {http://www.aaa.bb.cccc.dh/uploads/dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd.pdf},     urldate = {2011-03-02},     sortname = {Sun},     keywords = {www} } \end{filecontents}  \bibliography{bachelorarbeit_lit}  \begin{document} Test \cite{java96} \printbibliography \end{document}   edit  I don't know why but it still doesn't work. Is there an option to automatically break long URLs. I don't care where the break is but it should be done automatically. Even if I manually specify the "breaking character" it doesn't work (as the example shows). Why do gaps appear in my mesh when I animate it? As I attempt to complete a "Walk cycle" with my model, the mesh at the pants line separates from the skin as the model moves or as I rotate the controls. This happens in many places on my model and it reveals the model interior. It is a one piece mesh, though, so I don't understand why this happens. Is this caused by weight painting problems? How can I fix this? What is an unsigned char? In C/C++, what an unsigned char is used for? How is it different from a regular char? MySQL Date Between I will be traveling from Australia to Austria, transferring in Abu Dhabi. Is it alright for me to bring a vibrator in the check-in luggage, or are there laws against that?  It is approximately 17cm long. Are there any precautions to take? Sum from $n=1$ to $N$ of $\sin(nx)/2^n$ The question states:    I'm not sure what to do here. I used the geometric sum to do $2^{-n}$, but I can't think of a way to sum $\sin(n\theta)$ to get the required RHS.    Further working     Apologies for maybe being slow or missing something obvious here... I don't know how to get it into required form. The boundary of union is the union of boundaries when the sets have disjoint closures Assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$?   I can prove that $\partial (A \cup B)\subset \partial A\cup\partial B$ but for proving $\partial A\cup\partial B\subset \partial (A \cup B)$ it seems not trivial. I tried to show that for $x\in \partial A\cup\partial B$ WLOG, $x\in \partial A$ so $B(x)\cap A$ and $B(x)\cap A^c$ not equal to $\emptyset$ but it seems not enough to show the result. How to find out where an error occurs? Read a file line by line in reverse order