Here's what I'm tasked with showing: Let $(a_n)$ be a convergent sequence with $a_n\rightarrow a$ as $n\rightarrow\infty$. By the Algebraic Limit Theorem, we know that $(a_n^2)\rightarrow a^2$. Now prove this using the definition of convergence. In doing so, I have the following: Let $\epsilon&gt;0$ be given. Choose some $N\in\mathbb{N}$ so that for all $n\geq N$, $|a_n^2-a^2|&lt;\epsilon$. By algebra, $|a_n^2-a^2|=|a_n-a||a_n+a|$. Consider $|a_n+a|$. By the triangle inequality, $|a_n+a|\leq|a_n|+|a|$, thus $|a_n|$ is bounded by some $M\in\mathbb{N}$. I know I'm trying to choose $M$ so that $|a_n-a|&lt;\epsilon+M+|a|$. Where should I take it from here?
Question:  Let ${a_n}$ be a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$  Prove that  $\lim_{n\to\infty}a_n^2=a^2$  Attempt: Without using any properties of limits (this is a question in a section of the book that came before any list of any limit theorems)  By definition:  $|a_n^2-a^2|&lt;\epsilon$   $|a_n^2-a^2|$ $\leq$ $|(a_n+a)(a_n-a)|$ $\leq$ $|(a_n+a)| |(a_n-a)|$ $\leq$   $(|a_n | + |a|)(|a_n -a|)$ $&lt;$ $\epsilon$  And since I know that $\lim_{n\to\infty}a_n=a$, I'll subsitute that in for $|a_n|^2$ as $a^2$   $\therefore$ $a^2-|a|^2 &lt; \epsilon$ which is true since $a^2-|a|^2$ in my case.  I wonder if this is a possible solution?
What is the proper etiquette at Japanese onsen? Japan is a volcanically active country which means it has a plethora of onsen (hot springs) all around the country.  Visiting these onsen is supposed to be a popular vacation destination for the work-weary Japanese.    I've been told that, as with other things in Japan, there is a rather strict etiquette surrounding the use of the onsen.  In particular, the two things I've read about so far are   Swimming Suits are not allowed. It is very important that one bathe immediately before entering the onsen.   Are these true?  Are there any other taboos that a naive American should be aware of? What is the purpose of the EBP frame pointer register? I'm a beginner in assembly language and have noticed that the x86 code emitted by compilers usually keeps the frame pointer around even in release/optimized mode when it could use the EBP register for something else.  I understand why the frame pointer might make code easier to debug, and might be necessary if alloca() is called within a function.  However, x86 has very few registers and using two of them to hold the location of the stack frame when one would suffice just doesn't make sense to me.  Why is omitting the frame pointer considered a bad idea even in optimized/release builds? Formula to add minutes to date+time in Excel If I have a cell formatted as date+time in Excel, how can I add to this in terms of minutes? It would need to work whether I was adding 20 minutes, or 2,500 minutes.   So, for example if I had this spreadsheet, what formula could I put in B5 to get a result of 2013-09-22 09:10?    I tried using =B4+TIME(0,B3,0), but it gives 2013-09-21 09:10 which is a day off. It works fine though if B3 is less than a day's worth. What's the difference between using single and double quotation marks/inverted commas? Why does energy have to be emitted in quanta? I was reading some popular science book and there was this sentence. It says that energy has to be emitted in discrete portions called quanta. Otherwise the whole energy in the universe would be converted into  high frequency waves.  I'm not a physicist, so this conclusion seems to me like a huge leap.  First, we assume that energy is emitted in a continuous way (not in quanta).  And how do we get to the statement "the whole energy in the universe would be converted into  high frequency waves"? I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? I am very new to being able to use LaTeX to type up mathematical papers, and I just got my "system" set up. Currently I have installed TexWorks, and MikTex, and I have put together a 'hello world' article which seems to run well.   I have also used the syntax "\usepackage{amsmath}" and "\usepackage{amssymb}" in the beginning of my .tex file to write math symbols, only because some online tutorial said so.   My question is, firstly, are those the only packages I need to use for writing math, and secondly, I remember reading about some shortcuts(?) that can be used for writing math symbols, and/or user defined shortcuts that can be 'set' in the beginning so as to reduce the amount of LaTeX code needed to write mathematical symbols. However, I do not know if this is true, and if it is, how to go about 'installing' such a thing.   Does such a thing exist? Am I able to make my own shortcuts?  Thanks! I'm using the 1-25-19 version of 2.8 beta. You have to double tap "a" every time you want to deselect something which is very annoying. Is there a way to make "a" work the old way where "a" selects all if nothing is selected and "a" deselects all if something is selected? ie. no double tap and no alt? I wonder which of the two is correct:        I have a red and a blue car.   I have a red and a blue cars.      or        Everyone has a left and a right arms.   Everyone has a left and a right arm.      My (non-native) inclination would be for option 1, as it sounds better.  On the other hand, in the following examples, the plural sounds better to me.    Everyone has left and right arms. I have old and new cars. (when I only have only one old car and one new car) The left and the right arms are...   But are they correct? What is bracketing? I have seen this term a lot and I was wondering what, in basic terms, what bracketing/autobracketing is.  Also, why would one want to 'bracket'?  P.S. - there are other questions asking what DoF and WB bracketing is but none asking quite the same as this. Maybe I'm searching for the wrong thing, but I can't see how I can do this.  I have two servers:   Server A (has SQL Server 2012 installed) Server B (has SQL Server 2014 installed)   Server A provides an API via IIS. The code both inserts and gets data from the database depending on the API call.  Server B does exactly the same for its local database.  I would like to have it so that the two databases are consistent copies of each other. I do believe this is what HA accomplishes.   So a load balancer, for example, might direct API calls to A sometimes, and B sometimes, but the data needs to be consistent either way.  I found some documentation on HA for SQL Server 2012, but not on how to do this for a mixed environment of 2012 and 2014.  Any help or pointers to documentation would be great! Thanks. Why can two (or more) electric field lines never cross? The the title is self explanatory, I guess.  Why can two (or more) electric  never cross? Here is the top output which I gathered:    I noticed that top is showing vlc's cpu usage is > 100%.  Can anybody please justify me why it is showing like this, whether this is a bug in top application or something else. The relativistic energy-momentum equation is:  $$E^2 = (pc)^2 + (mc^2)^2.$$ Also, we have $pc = Ev/c$, so we get: $$E = mc^2/(1-v^2/c^2)^{1/2}.$$  Now, accelerating a proton to near the speed of light, I get the following results for the energy of proton:  0.990000000000000   c =&gt;    0.0000000011    J =      0.01    TeV  0.999000000000000   c =&gt;    0.0000000034    J =      0.02    TeV  0.999900000000000   c =&gt;    0.0000000106    J =      0.07    TeV  0.999990000000000   c =&gt;    0.0000000336    J =      0.21    TeV  0.999999000000000   c =&gt;    0.0000001063    J =      0.66    TeV  0.999999900000000   c =&gt;    0.0000003361    J =      2.10    TeV  0.999999990000000   c =&gt;    0.0000010630    J =      6.64    TeV 0.999999999000000   c =&gt;    0.0000033614    J =      20.98   TeV  0.999999999900000   c =&gt;    0.0000106298    J =      66.35   TeV  0.999999999990000   c =&gt;    0.0000336143    J =      209.83      TeV  0.999999999999000   c =&gt;    0.0001062989    J =      663.54      TeV  0.999999999999900   c =&gt;    0.0003360908    J =      2,097.94    TeV  0.999999999999990   c =&gt;    0.0010634026    J =      6,637.97    TeV  0.999999999999999   c =&gt;    0.0033627744    J =      20,991.10   TeV    If the LHC is accelerating protons to $7 TeV$ it means they're traveling with a speed of $0.99999999c$.  Is everything above correct? Thousands of 404 errors in Google Webmaster Tools Because of a former error in our ASP.Net application, created by my predecessor and undiscovered for a long time, thousands of wrong URLs where created dynamically. The normal user did not notice it, but Google followed these links and crawled itself through these incorrect URLs, creating more and more wrong links.  To make it clearer, consider the url     example.com/folder   should create the link     example.com/folder/subfolder   but was creating     example.com/subfolder   instead. Because of bad url rewriting, this was accepted and by default showed the index page for any unknown url, creating more and more links like this.     example.com/subfolder/subfolder/....   The problem is resolved by now, but now I have thousands of 404 errors listed in the Google Webmaster Tools, which got discovered 1 or 2 years ago, and more keep coming up.  Unfortunately the links do not follow a common pattern that I could deny for crawling in the robots.txt.  Is there anything I can do to stop google from trying out those very old links and remove the already listed 404s from Webmaster Tools? How to skip validation when a specific button is clicked? Abnormal PREEMPTIVE_OS_AUTHENTICATIONOPS Waits I would like to treat a user as a non-gender noun and refer to it with the gender-neutral pronoun, it. E.g.,     The user defines two variables, x and y. It then multiplies each variable by a prime number.    However, on  I found this:      The word "it", however, has an extremely impersonal connotation, even offensive, in common usage and is rarely used in English to refer to an unspecified human being or person of unknown gender. This is because the word "it" connotes that the person being specified is inferior to a person or is an object.   Is to appropriate to refer to a person of unknown sex as it?  Should I rephrase my sentence as follows:     The user defines two variables, x and y. The user then multiplies each variable by a prime number. Are if and whether equivalent in sentences like the ones below?     How to determine if my saddle is too high?   How to determine whether my saddle is too high?         We should check if everything is okay now.   We should check whether everything is okay now. A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?