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I'm trying to prove that the characteristic of any field $F$ is either $0$ or a prime number, but I have no idea what to do.  Help? Set of limit points of a subset of a Hausdorff space is closed. Let $X$ be a Hausdorff space and $A\subset X$.  Define $A'=\{x\in X\mid x\text{ is a limit point of }A\}$.  Prove that $A'$ is closed in $X$.  Relevant information:  (1.) Every neighborhood of a point $x\in A'$ contains a point $y\in A'$ distinct from $x$ (in fact, in a Hausdorff space, every neighborhood of $x$ contains $\infty$-many points of $A$ distinct from $x$)  (2.) In a Hausdorff space, every sequence has a unique limit.  On first glance, it should be an easy proof, but I've made little progress.  I was planning on showing $\overline{A'}=A'$.  Firstly, $A'\subset \overline{A'}$ trivially.  To show $\overline{A'}\subset A'$, proceed by contradiction.  Assume there exists $x\in\overline{A'}$ such that $x\not\in A'$.  This should yield an easy contradiction but I don't see it.  In particular, I'm unsure if the fact that $x\in\overline{A'}$ implies that there actually exists a sequence in $A'$ converging to $x$.  By (2) we know all sequences have unique limits, but do we know that elements in the closure are limits of sequences?  If this is true, it should yield an easy contradiction.  Any help? Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! Using the definite article before a country/state name The Punjab is a rich state.   Is it correct to use the before Punjab? Benefits and Drawbacks of Swapping Associations of Positivity and Negativity of the Proton and Electron For historical reasons, I understand that a proton's charge was assigned the label of "positive", the electron's charge was assigned the label of negative, and, crucially, equations of current in E &amp; M are written with the notion that current is the flow of positive charge despite the rarity of actual flows of protons in real systems (which I understand are mostly biological systems).   If the possibility existed to instantaneously and at no cost, retcon these associations such that protons have a negative charge, electrons have a positive charge, and current remains the flow of positive charge, would there be any beneficial or detrimental outcomes?  For instance, would there be:   new interpretations of physical laws that would provide different or better insights into any relevant subfield of Physics significant simplifications to notation in any field of Physics or Engineering significant increases in understanding in the fields of electricity, electronics, electrical engineering, radiation/radiative transport, etc. a noticeably easier learning curve for any subfield of physics dealing with the concept of charge   Additionally, is there a permutation of positive charge, negative charge, and flow of current definition that would improve upon the one I suggested? Why do residuals in linear regression always sum to zero when an intercept is included? I'm taking a course on regression models and one of the properties provided for linear regression is that the residuals always sum to zero when an intercept is included.   Can someone provide a good explanation for why this is the case? How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:  $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$  There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2  a + (n-1)\cdot d}{2}\biggr)$$   How do we prove the above two identities? Is it possible to create a "weak reference" in javascript? Is there any way in javascript to create a "weak reference" to another object?   Can anyone think of a way to implement this  behavior in javascript? How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? Assume two black holes in the most common size range, spiraling into each other until they merge. The event releases significant amounts of energy via gravitational waves, which warp the space-time.  If the distortion is powerful enough, it would get noticed in daily life, or perhaps even detected by the proprioceptors in the human body.  How close, as an order of magnitude estimate, would you have to be to the merger to perceive it like that, without any instrument? I'm level 16 which makes battling many Draugr Wrights hard. My defense is just way too low, and I don't have enough time or strong enough magic to take them out.   My one-handed skill is at 62 and my strongest weapon does 33+10 enchanted damage. Are there any ways to level up easily? I have been trying to install the latest version of java-8. However, I found out that the  due to changes in the java licence.  How should I update my java to 8.0_201 or 8.0_211 now?  Thanks! From SL Loney Trigonometry:     Prove that   $$r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$$   for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, and exradii $r_1$, $r_2$, $r_3$.   This one is a really dirty question.   I have tried a lot of stuff like $r=4R\times \Pi(\sin\frac{A}{2}$) and  $r_1=4R\times\sin\frac{A}{2}\cos\frac{B}2\cos\frac{C}2$ etc..  There are other ways like $-r+r_1+r_2+r_3=4R$,  but all these are way too cumbersome or clumsy.   Anything more intuitive? How To Get Notified When Hard Drives Fail In Dell PowerEdge Servers I have a Dell PowerEdge R415 and R515 and wondering how I can be alerted remotely when a hard drive fails in either server? Both are running RAID, so I need to be notified, so I can swap out the drive. Both servers only have the basic Baseboard Management Controller. Is this possible with a third party tool/software? Loop-cut won't go all the way around Where to place and how to read configuration resource files in servlet based application? If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. What is the variance of the weighted mixture of two gaussians? Say I have two normal distributions A and B with means $\mu_A$ and $\mu_B$ and variances $\sigma_A$ and $\sigma_B$. I want to take a weighted mixture of these two distributions using weights $p$ and $q$ where $0\le p \le 1$ and $q = 1-p$. I know that the mean of this mixture would be $\mu_{AB} = (p\times\mu_A) + (q\times\mu_B)$.  What would the variance be?    A concrete example would be if I knew the parameters for the distribution of male and female height. If I had a room of people that was 60% male, I could produce the expected mean height for the whole room, but what about the variance? What are the biggest pros and cons of  vs ? How to pop-up alert to a user when he/she navigate away from page without saving the Records in custom visual force page? How to prove that a number is irrational