Using the definition, that for every $\epsilon&gt;0$, there exists a $N\in\mathbb{N}$, so that $d(v_i,L)$ for all $i\geq N$, where $v_i$ is the $i$th vector in the sequence. And $L$ is the limit.  I assumed that let there be two limits, $L$ and $M$.  Now I choose my epsilon to be $\frac13d(L,M)$, but after that, how do I proceed? Plain algebra doesn't help.  Note that $d(a,b)$ is the distance between vectors $a$ and $b$ in $\mathbb{R^n}$.
I'm reading through a proof in my lecture notes:     Every convergent sequence of a metric space has a unique limit.   Suppose $\lim_{n \to \infty} x_n = l$ and $\lim_{n \to \infty} x_n = m.$ Then for every $\epsilon &gt; 0,$ there exists $N$ such that for every $n \geqslant N,$  $$d(l,m) \leqslant d(l,x_n) + d(x_n,m) &lt; \epsilon + \epsilon = 2 \epsilon. $$  So $(\star) \hspace{1mm} 0 \leqslant \frac{d(l,m)}{2} &lt; \epsilon \Longrightarrow \hspace{1mm} (\star^\prime) \hspace{1mm} \frac{d(l,m)}{2} =0 \Longrightarrow l = m.$  I can follow the proof up until the jump from $(\star)$ to $(\star^\prime)$. Why does $\frac{d(l,m)}{2}=0$?
Help with this trigonometry problem Prove the given identity.  $$\left(\sin\frac {9\pi}{70}+ \sin\frac {29\pi}{70} - \sin\frac {31\pi}{70}\right) \left(\sin\frac {\pi}{70}-\sin\frac {11\pi}{70} - \sin\frac {19\pi}{70}\right) =\frac {\sqrt {5} -4}{4}$$  Please help, I could not gather enough ideas, even on how to get to the first step. I thought of using the transformation formula (from sum to product) but did not find a fruitful result.   UPDATE 1.:  $\left(-\frac{1 +\sqrt{5}}{8} + \frac{1}{8} \sqrt{14\left(5 - \sqrt{5} \right)}\right)\left(-\frac{ 1+ \sqrt{5}}{8} -\frac{1}{8} \sqrt{14 \left(5 -\sqrt{5} \right)}\right) $,    I noticed that Alexis had got to this point.  Can anyone explain me how did he get here, with calculations. Please help.  Thanks in Advance. Statistical significance (p-value) for comparing two classifiers with respect to (mean) ROC AUC, sensitivity and specificity I have a test set of 100 cases and two classifiers.  I generated predictions and computed ROC AUC, sensitivity and specificity for both classifiers.  Question 1: How can I compute p-value to check if one is significantly better than the other with respect to all scores (ROC AUC, sensitivity, specificity)?    Now, for the same test set of 100 cases, I have different and independent feature assignments for each case. This is because my features are fixed but subjective and provided by multiple (5) subjects.  So, I evaluated my two classifiers again for 5 "versions" of my test set and obtained 5 ROC AUCs, 5 sensitivities and 5 specificities for both classifiers. Then, I computed the mean of each performance measure for 5 subjects (mean ROC AUC, mean sensitivity and mean specificity) for both classifiers.  Question 2: How can I compute p-value to check if one is significantly better than the other with respect to mean scores (mean ROC AUC, mean sensitivity, mean specificity)?    Answers with some example python (preferably) or MatLab code are more than welcome. What are the differences between $$, \[, align, equation and displaymath? If I have a PATH variable that (when exploded onto multiple lines) contains something like this (with a few notes added by me):  /Users/brandon/.rvm/gems/ruby-2.0.0-p353/bin /Users/brandon/.rvm/gems/ruby-2.0.0-p353@global/bin /Users/brandon/.rvm/rubies/ruby-2.0.0-p353/bin /opt/local/bin /opt/local/sbin /usr/local/sbin /usr/local/bin   # &lt;&lt;--- notice this path appears again later /usr/bin         # &lt;&lt;--- notice this comes AFTER /usr/local/bin /bin /usr/sbin /sbin /usr/local/bin   # &lt;&lt;--- here it is again! /usr/local/git/bin /Users/brandon/bin /Users/brandon/.homebrew/bin /usr/local/mysql/bin /usr/local/share/npm/bin /Applications/Postgres93.app/Contents/MacOS/bin /Users/brandon/.rvm/bin   is it possible for the presence of the second /usr/local/bin/ to cause the /usr/bin to take precedence over /usr/local/bin?  This seems crazy (to me: I've never heard of such a thing) but that's exactly the behavior I'm seeing.  I've just installed PHP with Homebrew, and yet the system PHP (from /usr/bin) is taking precedence:  $ which php /usr/bin/php   BUT, after manually altering the PATH (rather than digging through all the files sourced by my .zshrc to try to figure out why &amp; where the duplication is coming from) to remove the second instance of /usr/local/bin, I find that /usr/local/bin is taking precedence over /usr/bin as I would have expected it to in the first place:  $ which php /usr/local/bin/php   What would cause this?  Is this normal? Is it possible to do a cross page POST to an ASP.NET, from a static HTML form? How to get my hover-menu back? Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$.   I am reasoning for the absurd and I am assuming that $gcd (x ^ 5, x ^ 6) = ax ^ 2 + bx ^ 3\in\mathbb{Q}[x^2 ,x^3 ] $, then, for the Bezout identity there exists $p (x) = cx ^ 2 + dx^3$ and  $q(x)= ex ^ 2 + fx ^ 3$ in $\mathbb{Q}[x^2 ,x^3 ]$ such that $x ^ 5p (x) + x ^ 6q (x) = ax ^ 2 + bx ^ 3$, which is absurd since $$9 = deg (x ^ 5p (x) + x ^ 6q (x)) \neq  deg (ax ^ 2 + bx ^ 3) = 3.$$  Is it okay what I did? Thank you very much. TL2018 (MacTeX) and TikZ calligraphy library Merge flash site into one executable ffmpeg - how to apply filters without losing quality Optimum Bone To Water Ratio For Pork and Beef Stocks I am tinkering with my stock recipes.   For pork stock I have been using 8 cups of water over a roasted trotter, a roasted backbone section with about 4 inches of rib attached, and a 1/4 inch slab of roasted pork belly.   For the beef stock I have been using 8 cups of water over an equivalent amount of roasted beef bones (knuckles and backbone/rib).   This gives a very flavorful stock and, for cost reasons, I want to determine how much I can reduce the concentration (by adding more water) without reaching too watery a taste.   Does anybody have any guidance on this topic? Phone keeps rebooting on Volume Up + Power (trying to install ROM) I tried to install Android's ROM and got ERROR 7, then I wiped out cache, data, internal storage, where was twrp.apk installed. After that I cant get into boot menu (with Install, Wipe... buttons). Then I copied twrp to the phone: sudo fastboot flash recovery &lt;twrp.img&gt; Tried versions 3.2.1-0 and 3.4.0-0, because I heard for certain phone (I have Xiaomi Mi4) it's necessary to use proper version, though I don't know which. What's going on now: I can successfully can get to the nice splash screen using Volume Down + Power, then on my PC I give sudo fastboot reboot command and press Volume Up + Power and here is the problem: It has no effect, it just keeps rebooting (I see only Xiaomi's logo). I have 3 folders with large amount of data in each which has to be maintained seperately. Can I create a sym link or shortcut.in windows which should show me the merged contents of the 3 folders. I am not looking for library conf in Win7.   Any suggestion is highly appreciated. Cycles final render pitch black? In solid mode with Material or Texture, it is shown fine.   But in the Render mode, everything is pitch black. I have a sun, and the Opacity of the scene is set to 100%.    I am new to using Blender, so I have no idea what is wrong. If you could help, I would appreciate it. If you need more information to help me, I would be glad to give it. What is the correct form in the following sentences, "both" and "both of them"?      a) "Both of them are unique."      b) "Both are unique."   If "both" by itself already contains "both of them", so I can conclude that the first option is not correct. Isn't it? The correct omission of the definite and indefinite articles When he was just a professor, Dumbledore taught Transfiguration. As Headmaster what was he doing all day? In the 6th movie he was diligently looking for Horcruxes, but what about the other times?  When Fred and George pass the Maurader's Map to Harry they note that Dumbledore is pacing in his office. "He does that a lot" one of them says. Did Dumbledore just endless pace in his office as Headmaster, afraid to leave it for fear someone would know that he produced no work?   Sure, every year he had to go find a new DADA teacher, but most of those it seemed he did through correspondence. Horace Slughorn was the only one that seemed to resist. Dumbledore's 6th year may have been the only one where he earned his paycheck. What are turret connectors? While working on a project recently, I noticed that Digi-key has a product category . I've never seen connectors like this before, and they don't look like any kind of contact I've seen before. What does the mating connector look like? I can't envision any mating contact that makes any sense for these.  On top of that, they look like they have to be individually machined, which can't be a cost-effective process. Any smart ideas on finding the area of this shaded region? How to make a file (e.g. a .sh script) executable, so it can be run from a terminal