how to prove this inequality by real anlysis  $\frac{|x+y|}{1+|x+y|} \leq \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}$  I finally came with the following inequality. but I dont know what to do after that  $\frac{|x+y|}{1+|x+y|} \leq \frac{|x|}{1+|x+y|} + \frac{|y|}{1+|x+y|}$
Prove that for every $a, b\in R\setminus\{0\}$ is correct this inequality:  $$\frac{|a+b|}{1+|a+b|}&lt;\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}.$$
Is there a limit to the number of magic rings a character can wear on one hand? Difference between $\Delta$, $d$ and $\delta$ Is this a correct demonstration for why elements above untriseptium cannot exist? With the confirmation that elements 113, 115, 117, and 118 are indeed fundamental elements that are now to be named on the periodic table, the next question is: what is the highest atomic number possible for an element? Feynman had a go at this years ago, and he derived (according to my limited understanding) that above element-137 (informally denoted 'Feynmanium') the electrons in the nearest orbit about the nucleus would be traveling at a velocity greater than the speed of light. (Note I am considering the Bohr model of the atom in this question.) I wished to demonstrate to my friends why this is and I came up with the following explanation.  According to Bohr's quantum condition, the angular momentum of an electron about the nucleus is given by $$ L=m_evr_n=n\hslash \implies r_n=\frac{n\hslash}{m_ev},$$ where $m_e$ is the mass of the electron, $v$ its velocity, $n$ an integer, and $r_n$ the radius of the $n$th possible orbit. Because we are concerned with the nearest orbit to the nucleus, $n=1$; and thus $$r=\frac{\hslash}{m_ev}.$$   Now, according to Coulomb's law, if the electron orbits about the nucleus the centripetal motion can be described by $$\frac{Ze^2}{4\pi\epsilon_or}=m_ev^2, $$ where $Z$ denotes the number of protons in the nucleus (the atomic number), and $e$ the elementary charge. Solving for $Z$ and substituting in $r$ from above yields $$Z=\frac{4\pi\epsilon_o \hslash v}{e^2}. $$ But what is $v$? Well, the maximum velocity an electron could ever have is the speed of light, and we wish to find the atomic number associated with an orbiting electron traveling at this speed, so we set $v=c$ and obtain our final result of $$Z=\frac{4\pi\epsilon_o\hslash c}{e^2}\approx 137.521,$$ which implies that for $Z&gt;137$, the electrons at a position of $n=1$ in the Bohr model would have a velocity $&gt;c$; and thus the highest atomic number achievable on the periodic table is 137.  Again, I just want to make sure this is a correct method for deriving element-137 before presenting. Perhaps one could explain how relativity plays a role here. I know Feynman used the Dirac equation to get this result...so could anyone (subsequently of course) expatiate on this in a simplistic manner? Thanks! Is 1+1 =2 a theorem? A theorem is defined to be a mathematical statement that is proven to be true. The statement $1+1=2$ has definitely been proven in the history of mankind (Russel and Whitehead had once proven it in the book Principia Mathematica).   So can it be considered as a theorem? What determines something to be a theorem (besides it being proven to be true)? The  feat (PHB, p. 168) says:     While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.   The  feat (PHB, p. 170) says:     When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.   On this very topic, Mearls has :     Q: Do Polearm Master and War Caster combine to allow a magic user to make a spell opportunity attack when they enter reach?   A: No - polearm master applies only if you use the weapons it lists to make the attack     Another question on this very topic:   But since then, we now have spells like  (SCAG, p. 142):     As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.    So, with the addition of booming blade, if I have a quarterstaff + War Caster + Polearm Master, could I use my reaction to cast booming blade if someone steps into my reach? How did they post process before Photoshop? Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$ Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions Let's $X$ and $Y$ have CDF functions admitting moment of order $1$. Let's be $F$ cdf of $X$ and $G$ cdf of $Y$.  I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(u)-G^{-1}(u)\mid du\,.$$ Object Positions Reset On Blender Load Double dual mappings $\newcommand{\Hom}{\operatorname{Hom}}$Let $U,V$ be vector spaces. Denote the dual space of $U$ with $U^\intercal$ and the dual mapping of a linear map $\Phi$ be $\Phi^\intercal$. Define the double dual embedding on $U$ to be $\iota_U:U \to U^{\intercal\intercal}$. ($u \in U, f \in U^\intercal$) $$ \iota_U(u)(f) = f(u) $$     If $\Phi \in \Hom(V,U)$, then $\Phi^{\intercal\intercal} \circ \iota_V = \iota_U \circ \Phi:V \to U^{\intercal\intercal}$   Proof: Let $v \in V, f \in U^\intercal$. Then $$ \iota_U \circ \Phi(v)(f) = f(\Phi(v)) $$ but how do I show that $$ \Phi^{\intercal\intercal} \circ \iota_V(v)(f) = f(\Phi(v)) $$ ? Efficient way to find $[\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{2}): \mathbb{Q}(\sqrt{3}, \sqrt{2})]$ I want to show rigorously that this is 2. I'm sure there's a faster way than by trying to see if $\sqrt{5} = a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$ (linear combination of the basis elements for $\mathbb{Q}(\sqrt{3}, \sqrt{2})$). I'm just not sure what. In H. P. Lovecraft's The Case of Charles Dexter Ward, Dr. Willett      Enters Joseph Curwen's dungeon and resurrects a dead person identified as No. 118 by reciting the incantation of the Ascending Node over his salts.   Dr. Willett is told by      The resurrected Joseph Curwen, impersonating Charles Dexter Ward, who he secretly murdered,   that   " 'twas Number 118, and I conceive you would have shook had you looked it up in my list in t'other room."  No. 118 himself gives Dr. Willett a note asking the doctor to      kill Joseph Curwen.   It soon becomes clear that No. 118 is opposed to the nefarious dealings that have been transpiring and desirous of being an ally of Dr. Willett.  Who is No. 118?  Earlier in the story,      The grave of Ezra Weeden, a primary instigator in Joseph Curwen's initial downfall and the romantic rival of Curwen, is robbed. There doesn't seem to be any other mention of Weeden in the 20th century occurrences, so without a further connection the passage seems superfluous.   The shared goals of these two people seem to indicate that No. 118 may, in fact, have been that person, but the      Note given by No. 118 to Dr. Willett in eighth century Latin seems to indicate someone much older than Ezra Weeden.   It also occurred to me that No. 118 could be      The recently murdered Charles Dexter Ward himself. Ward's reference to killing Dr. Allen (who is actually an alias of Curwen) and dissolving his body with acid and No. 118's mention of doing the same to Curwen seem to support the idea, but that still doesn't explain the archaic note. Ward is described as writing in a somewhat modern style, at least as opposed to Curwen. It also doesn't explain why Curwen would want to murder Ward if he was going to resurrect him later anyway. "To be subject to" vs. "to be subjected to" Trying to find the original story (I read it in Russian translation, original was most likely English unless it was Lem).  The story plot had artificial birds designed to serve as cops. The birds started attacking humans who were percieved as being aggressive towards anything alive. So they created more advanced birds to protect against the first ones. This worked for a while until the second kind started attacking humans as well. It ended with the idea that they would need another generation to attack the second-generation birds.  The story wasn't very long. I read it a while back (mid-1980s, which means the original was written even earlier, probably before 1980).  Despite a somewhat Asimov-like "misbehaving robots" topic, I'm pretty sure it wasn't Asimov. Are the "umbles" etymologically related? From      And we show telltale signs of early hypothermia, what the doctor and   the National Institutes of Health and other cold-weather cognescenti   call the "umbles."      First we lose some fine motor skills, McLaren says, "and we start to   fumble. And tumble. And stumble."      Then our cognitive abilities are compromised. "We mumble and we   grumble."   Are the "umbles", such as fumble, tumble, stumble, mumble and grumble, etymologically linked, either in terms of sharing a common ending, or entire words having a common ancestor?  Trying to do a search of the etymology of the umbles ended up giving the etymology of the word  instead. How do I clear the render border? In the 3d viewport I am using a border region to only render things inside the selected region. I did this with CtrlB. I cannot get rid of the region. How can I make the border disappear? Why did Sirius Black's family live among Muggles? Usually, when we see two letters put together it is to define the pronunciation or to differentiate synonyms or just its foreign origins. But some words don't seem to have any reason to double up.     Why does vacuum have two of the letter U?  Why does aardvark have two of the letter A?  Why does llama have two of the letter L? Gauss law and electric flux Can you please explain me what is electric flux. I know it is $\int\mathbf{E}\cdot\mathrm{d}\mathbf{S}$, but in many place its given flux means to flow. But how does electric field "flow"? Also if we assume its the electric field lines (can you also specify what exactly are these) are "flowing" how does $\int\mathbf{E}\cdot\mathrm{d}\mathbf{S}$ give the total electric field lines passing through that area? (can you please elaborate when you explain?)  Pls. I know it shouldn't come here but there is no page for electric flux. Illustrator: how to change the artboard number?