Is this following characterization true?  Let $x_n\subseteq \mathbb{R}$ a sequence, and define $\limsup_nx_n:=\lim_n\sup_{m\geq n}x_m$. Then $\limsup_n x_n=\max\{L$ such that $L$ is the limit of a converging subsequence$\}$  My try (for $x_n$ bounded): yes, indeed if $x_{n_k}$ is a convergent subsequence we have $\limsup_n x_n=\lim_n \sup_{m\geq n}x_n=\{$ this limit exists, hence it is equal to the limit of any converging subsequence $\}=\lim_k\sup_{m\geq n_k}x_m\geq \lim_k \sup_{l\geq k}x_{n_l}=\lim_kx_{n_k}$. Here, all the sequences inside $\lim$ are converging since monotonous and bounded. Also, from the original definition of $\limsup$ it is clear there is a converging subsequence $x_{n_k}\rightarrow _k \limsup_nx_n$, so that my claim follows. Is this (also?) true for unbounded sequences?
I have come across these three definitions of the limit superior (or upper limit) and the limit inferior (or lower limit) of a sequence of real numbers and I wonder how to establish the equivalence of these.    Walter Rudin: PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition:  Definition 3.16: Given a sequence $\{s_n\}$ of real numbers, let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_k} \to x$ for some subsequence $\{s_{n_k}\}$. This set contains all subsequential limits ..., plus possibly the numbers $+\infty$, $-\infty$. We now ... put  $$s^{*} = \sup E,$$ $$s_{*} = \inf E.$$ The numbers $s^{*}$, $s_{*}$ are called the upper and lower limits of $\{s_n\}$; we use the notation  $$ \lim_{n \to \infty} \sup s_n = s^{*}, \, \, \, \lim_{n \to \infty} \inf s_n = s_{*}.$$ Tom M. Apostol: MATHEMATICAL ANALYSIS, 2nd edition:  Sec. 8.3: Definition 8.2: Let $\{a_n\}$ be a sequence of real numbers. Suppose there is a real number $U$ satisfying the following conditions:    i} For every $\epsilon &gt; 0$ there exists an integer $N$ such that $n &gt; N$ implies $$a_n &lt; U + \epsilon.$$ ii) Given $\epsilon &gt; 0$ and given $m &gt; 0$, there exists an integer $n&gt; m$ such that $$a_n &gt; U - \epsilon.$$ Then $U$ is called the limit superior (or upper limit) of $\{a_n\}$, and we write $$U = \lim_{n \to \infty} \sup a_n.$$ Statement (i) implies that the set $\{a_1, a_2, a_3, \ldots \}$ is bounded above. If this set is not bounded above, we define $$\lim_{n\to\infty}\sup a_n = +\infty.$$ If the set is bounded above but not bounded below and if $\{a_n\}$ has no finite limit superior, then we say $\lim \sup_{n\to\infty} a_n = -\infty$. The limit inferior (or lower limit) of $\{a_n\}$ is defined as follows: $$\lim_{n\to\infty}\inf a_n = -\lim_{n\to\infty}\sup b_n,$$ where $b_n = -a_n$ for $n= 1, 2, 3, \ldots$.    Robert G. Bartle and Donald R. Sherbert: INTRODUCTION TO REAL ANALYSIS, 3rd edition: Exercises for Section 3.3: Problem 10: Let $(x_n)$ be a bounded sequence of real numbers, and for each $n\in \mathbb{N}$ let $s_n \colon= \sup \{x_k \colon k \geq n\}$ and let $t_n \colon= \inf \{x_k \colon k \geq n\}$. Prove that $(s_n)$ and $(t_n)$ are monotone and convergent. Also prove that if $\lim (s_n) = \lim (t_n)$, then $(x_n)$ is convergent. [One calls $\lim (s_n)$ the limit superior of $(x_n)$ and $\lim (t_n)$ the limit inferior of $(x_n)$.]   Now how can one show that the above three definitions are equivalent (i.e. these three definitions are of the same pair of numbers)?
If a permanent turns into a(nother) creature, has it entered the battlefield? I was playing a game last night with a guy who was playing a new Simic Gatecrash deck, and he claimed that if he made a land into a creature, it counted as entering the battlefield, thus triggering evolve (or other "enters the battlefield" abilities).  I thought this was incorrect, being that the creature didn't actually enter the battlefield, it was just created.  Which of us was right? $$\lim_{x \to 0} \frac{f(x)}{|x|}=1$$ Then prove that $f$ is not differentiable at $0$. I was given this assignment and I'm not sure about my proof.. it feels like I'm assuming too much and not explaining with enough detail. Is there anything I need to add to make it sufficient? Assume toward contradiction that $f$ is differentiable at $0$. Then by definition we know that $f$ is continuous at $0$. Therefore $\lim _{x\to 0}f\left(x\right)=f\left(0\right)$ We know that $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}=1$ Therefore we can conclude $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \left|x\right|=1\cdot 0=0$ But we can also rewrite this as $\lim _{x\to 0}\left(f\left(x\right)\right)\cdot \frac{\left|x\right|}{\left|x\right|}=f\left(0\right)\cdot 1=0$ Therefore $f(0) = 0$. Since $f$ is differentiable at $0$ we have $\lim _{x\to 0}\frac{\left(f\left(x\right)-f\left(0\right)\right)}{x-0}=\lim _{x\to 0}\frac{\left(f\left(x\right)-0\right)}{x}=\lim \:_{x\to \:0}\frac{\left(f\left(x\right)\right)}{x}\cdot \frac{\left|x\right|}{\left|x\right|}=\lim \:\:_{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\frac{\left|x\right|}{x}$ Since the one sided limits of $\frac{f\left(x\right)}{\left|x\right|}$ exist, and are both equal to 1, we have $\lim \:_{x\to 0^+}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^+}\frac{\left(f\left(x\right)\right)}{x}\cdot \:\:\:\frac{x}{x}=1\cdot 1=1$ $\lim \:_{x\to 0^-}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^-}\frac{\left(f\left(x\right)\right)}{-x}\cdot \:\:\:\frac{-x}{x}=1\cdot -1=-1$ Since the limit doesn't exist, we have a contradiction to the fact $f$ is differentiable at $0$. Therefore $f$ is not differentiable at 0. Why don't we consider jerk in physics classes? When I got more into physics, I started asking myself if just like acceleration represents the growth of speed, something else could also represent the growth of acceleration itself. And it came that is exists and is called jerk. Before I thought about this, I thought that acceleration was the highest derivative that made sense on physics.  My question is, why can we perform accurate calculations without account for jerk? I never needed it in a single physics class! I've also read that it is seldom used.  Also, while position, speed and acceleration seem familiar to me due to everyday experience, why doesn't jerk seem familiar to me? I never tell someone when driving a car "you're jerking too much", whereas I could say "you are accelerating too much".  Is it that jerk has little influence because it is almost constant? Even if it is not, what about higher order derivatives? Why don't derivatives higher than acceleration have much influence on calculations? We perform a lot of reasoning in everyday life without resorting to it! (I am not asking how to block my posts on my timeline.)  I am asking how I can block my activity, such as posting in open groups, making comments on people's posts, adding a friend, liking a photo, liking a post, etc., from people seeing it in their news feeds?  I don't want people seeing my interactions with other people. Daily Ubuntu base update I am getting Ubuntu base update almost every day. This seems strange to me. What am I missing?   (I am using Ubuntu 14.04 LTS) Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$ So I am trying to prove the above stated formula. I am completely new to number theory so I thank you in advance for baring with me. I start off by using the definition:  $\frac{\phi(n)}{n} = \sum_{d|n} \frac{\mu(d)}{d}$  Where $\phi$ is Eulers function, and $\mu$ is the Mobius mu function. From here, I use the Mobius inversion formula to give:  $\frac{\mu(n)}{n} = \sum_{d|n}d\frac{\phi(\frac{n}{d})}{n}\mu(d)$  Now summing $n$ up to $x$:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \sum_{n \leq x}\sum_{d|n}d\frac{\phi(\frac{n}{d})}{n}\mu(d)$  Now I'm not fully sure on how I can rearrange things inside the two summations on the right hand side, so leaving things as they are; using that $\sum_{d|n} \mu(d) = 1$, if and only if $n =1$. Hence:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \sum_{d|1}d\phi(\frac{1}{d})\mu(d)$  Thus $d=1$. Subbing this in:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \phi(1) = 1$  I am not sure what I have missed (or done wrong) as to have an equality and not a triangle relation. Any suggestions or ideas would be greatly appreciated Change cursor color after log out on ubuntu 18.04.2 LTS I've set my cursor color on gnome-tweaks to be DMZ-Black, which works fine when I log in and lock the computer. But, after I log out, the cursor color changes to white.  Is there a way to change the color of the cursor after log out?  My gnome-tweaks setting: Determining true mean from noisy observations Continuous probability distribution with no first moment but the characteristic function is differentiable I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere.  Cauchy distributions do not fulfill this, as their characteristic functions are not differentiable at $0$. Does anybody here have an example? As far as I know, current division for spesific place's formula is: Ix=(Req/Rx)*Isource   However, here,the solution is 3/(3+2+4)*6. How is this a current division ? Can you explain this to me Proof that the irrational numbers are uncountable Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable? What are some good introductions to philosophy? I've been searching the questions posed here, and don't seem to find one that gives the answers I am looking for. I've proposed to start a  on meta but no one proposed anything, so I feel I must ask this question: What are some books generally regarded as good for a general introduction on philosophy? By this I mean good books that can introduce you to philosophy in general and throughout the times, not to some philosophy or period in particular.   almost answers mine, except for the fact that most things proposed there are not books, thus not giving me much options to weigh upon.  regards contemporary philosophy in particular. And  has the same problem as the first one, plus the fact that it asks for layman's terms, which is not what I am looking for.  I've been tempted to buy Will Durant's , but am not sure if it is the best book to get started on philosophy (plus, the book has some mixed reviews). I also checked , which seems to be unanimously good, but has the format of dictionary/encyclopaedia, which is not exactly what I am looking for (at least as a main source).  also seems to be a good introduction regarding how to do philosophy, rather than to its history and ideals (this also interests me, but I'd rather learn about the philosophers and their theories first).   It is also important to point out that I have little notions (or none at all) on philosophy. I would like to read something that covers from classical to contemporary philosophy generally (although I do not know if there is such a book), so that from there I could pick some author, period or branch of philosophy in particular to read further on. So, to sum up, I would like to 'hear' your opinions on what are some books generally regarded as good for a general introduction on philosophy. If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Mathematics Ph.D program foreign language requirement I will soon be attending a Ph.D program in mathematics. I do not know how common this practice is, but the department has a requirement that graduate students must learn a foreign language and be able to read and write mathematics in that language. The three choices are French, German, or Russian. I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career. Thanks in advance in any response. I love the subset of collective nouns known as the terms of venery. These are collective nouns specific to a particular group of animals. Some of the more inventive examples are: a murder of crows, a crash of rhinos, a mischief of mice, and a puddling of ducks (specifically swimming ducks).  Is there a standard for these group names and is there a central reference for them as with other zoological taxonomy?  I can see in the Wikipedia articles that the practice has its origins in Medieval Hunting tradition, but the Kangaroo (Mob), for instance would have been unknown at that time, so how do new collections come into being? Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations? Is it ok to have mixed grounds and neutrals on bars in a breaker box? So to continue my line of electrical questioning (and perhaps narrow down my ), I took a look through the two panels in this house.  There is a 200amp main service in the basement, which then feeds up to a 100amp sub-panel for the main floor.  This 100amp sub feeds a kitchen (fridge, microwave, dishwasher, gas range), a bathroom, 3 bedrooms, and a living room.  The 200amp main feeds the 100amp sub, 2 bedrooms, a living room, a washer/dryer (gas dryer), a utility room (well pump, pressure tank, gas water heater, gas furnace), and an A/C.  My first concern is that the main service panel has the bare ground and neutral wires mixed on the two bus bars.  Reading around, some say this is OK, other's say it is bad.  Any thoughts?  The sub-panel is wired with grounds and neutrals on separate bars.  My second concern- is a 100amp sub sufficient for the main floor? Out-of-control starship results in some crew members landing in a rebirthed universe? What do I do if I don't have a META-INF folder or minecraft.jar? I have looked all over the Internet for solutions but I have got nothing.  I want to install the minepainter mod but I need a META-INF folder and a minecraft.jar.