Let $(X,d)$ a metric space. Show that $d'(x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ define a metric on $X$.  Hello, I have this problem. The triangular inequality is my difficulty. Indeed, we have that show that $d'(x,z)\le d'(x,y)+d'(y,z),\forall x,y,z\in X.$ Good, I do this...  $d'(x,z)=\dfrac{d(x,z)}{1+d(x,z)}\le \dfrac{d(x,y)+d(y,z)}{1+d(x,z)}=\dfrac{d(x,y)}{1+d(x,z)}+\dfrac{d(y,z)}{1+d(x,z)}.$  I don't understand how does with $d(x,z)$? Or if exists any form for bounded this expression.
Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$  Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$  b) show that $ d$ and $ d_b $ are equivalent metrics. c) let $(X,d) $ be $(\mathbb{R}, |\cdot|)$ Show that there exists no $ M&gt;0$ such that $ |x-y| $ $\leq$ $Md_b$ $(x,y)$ for all $ x,y$ $\in$ $\mathbb{R}$  I have calculated $f(t)$ and $f'(t)$ and from this I know $f(t)$ is an increasing function as $f'(t)$ is strictly positive. But I don't know where to go from here or how to do parts b) and c)
Using separation of variables to solve the wave equation I'm trying to teach myself separation of variables and have been following some notes for the wave equation, but there's one part which really confuses me and I'm not exactly sure how it makes the step.  For the wave equation   $$u_{tt} - c^2 u_{xx} = 0$$   with length $l$ and fixed ends, $u(0,t)=u(l,t)=0$ we seek a solution in the form $$u(x,t)=v(x)q(t)$$  and substituting this into the equation gives $$\frac{1}{q(t)}\frac{d^2 q}{dt^2}=c^2 \frac{1}{v(x)}\frac{d^2 v}{dx^2}=-\omega ^2$$   and I understand all of this so far. But when solving the equation  $$q''+\omega ^2 q = 0$$ the general solution is $$q(t)=A \cos(\omega t +\alpha)$$ where $A$ and $\alpha$ are constants - which is the part which I'm not exactly sure on how it gets to.   Could someone explain this step to me? Thanks! I'm using biblatex for my references.  Normally, if I use \cite{source1, source2, source3}, I get a citation like  [3, 45, 98]   If I now want to give a page number for one of them, it looks confusing imho, as I should get (did not try it out up to now, honestly)   [3, 45, 78, p. 23f, 98]   So imho it would make sense to change one of the separator from a comma to e. g. a semicolon like:  [3; 45; 78, p. 23f; 98]   Question 1: Does that make sense, or are there other conventions for citing sources with and without page numbers together?  (Up to now I collected the citations without page number in one cite command and added singular \cite[][p. xy]{} commands for those with page numbers., so I'd get:   [3, 45, 98][78, p. 23f]   )  the \cites[]{}[]{}[]{} could also do that, but it also uses the same separators for entries and postnotes.  Question 2: How could I change the separator globally for those number-citations from comma to semicolon in biblatex? "Recent Badges" list is displayed without margins on "Home" page When the &quot;Hide Hot Network Questions&quot; preference is enabled (i.e. HNQ sidebar is hidden), we see &quot;Recent Tags&quot; and &quot;Recent Badges&quot; in the homepage instead of the HNQ. However, the &quot;Recent Badges&quot; list is displayed with no margins. ex.)   I think that the appropriate margins were set in the past. cf.) Is it possible to take screenshots on the 3DS? In Pokemon X and Y there are specific locations in the game that allows you to save what are essentially 3D screenshots from the upper screen using a virtual camera. This made me wonder - is it possible to take screenshots, 3D or otherwise, from within the 3DS? Where are minecraft languages stored and how can I add a new one? What does "Warning: untrusted X11 forwarding setup failed: xauth key data not generated" mean when ssh'ing with -X? Space ship manned by only AI returns home to a different earth, back in time I read this short story 30 years or more ago.  A space ship is returning home to Earth from a battle. There are no people on board this ship, only an AI and the ship's computer. When the ship tries to contact Earth it gets no immediate response. The ship attempts to make contact for some time and then is successful with Houston Control (I think).  The AI discovers that it is on a different Earth from the one it is from and that it has also traveled back in time. (The enemy the fleet that it fought still exists).  The space ship comes from an Earth were the Greek empire did not get destroyed.  In the end the AI crashes the ship on the Capitol and gets destroyed, but the computer survives so that the human race has the knowledge of how to build ships and defend against the enemy.  Does anyone know the name of this story? For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere I want to show that for $f: \mathbb{R} \rightarrow\mathbb{R}$ Lebesgue integrable  $\displaystyle{\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere.  It is clear to me that this statement is true and I have tried to prove it by contradiction, by assuming that either $\displaystyle{\lim_{n \to +\infty} f(x+n) =\infty}$ or$\displaystyle{\lim_{n \to +\infty} f(x+n) =K}$ with $K\in \mathbb{R}$ .  In the first case  we can follow that $|f|=\infty$ so that $\int_\mathbb{R}f=\infty$ and we have a contradiction.  In the other case we know that there must be an $a\in \mathbb{R}$ so that $\forall x&gt;a$ we have $f_+(x)&gt;K/2$ and by that:  $\int_{{\mathbb{R}}+}|f|&gt;\int_{\mathbb{R}_+}f_+=\int_{[0,a]}f_++ \int_{(a,\infty)}f_+&gt;\int_{[0,a]}f_++\int_{(a,\infty)}K/2=\infty$  Would this be enough to show the statement? Can I even make the assumption that the limit of $f(x+n)$ "exists"? Any hints would be great. Capturing speaker output in Java How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$? For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$:   Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum.  I've shown that case $1$ implies $7\mid a$ and $7\mid b$, so I just have show that case 2 isn't possible - I'll be happy for any input.  MfG, Karl How to make Portable QGIS 3 Is it possible to make QGIS 3 portable with changed location for Qgis setup parameters folder.  For QGIS 2 it looked like this in qgis.bat file:     qgis --optionspath "X:\qgis" --configpath "X:\qgisini"   For QGIS 3 it doesn't work. Setup parameters are default placed in:      C:\Users\xxx\AppData\Roaming\QGIS A singularity, by the definition I know, is a point in space with infinite of a property such as density.  Density is Mass/Volume.  Since the volume of a singularity is 0, then the density will thus become infinite because Mass/0 = undefined  However, is it possible to have a singularity with a mass of 0? 0/0 is indeterminate, but would it be possible for a singularity to exist even if its mass and volume were zero? Fight 707 arrives at/in Washington D.C. in ten minutes Fight 707 arrives at Washington D.C. in ten minutes. Fight 707 arrives in Washington D.C. in ten minutes.     Do we have to use 'at' or 'in' in this case? Washington D.C. is a big city, so we have to use 'in' normally. However, because the plane leaves the city soon, the city can be a particular point on a map. So, can we use 'at' in this case? Approximate order statistics for normal random variables When and why is the [featured] tag removed by the Community user? I've found that the Meta (moderator only) tag  is automatically  by the  user. So, I want to know when and why is the tag  is removed by this user?  Is there any mechanism for this removal or is there simply a specified time period after which the  is automatically removed by the Community user? Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially)   It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone arranged to height from shortest to highest, but it doesn't hold up for the LHS.   Alternatively, trying to visualize the LHS, I noticed that it's like a right angle tetrahedra:  1  2!+2!  3!+3!+3!  ...  But it doesn't help to see a connection to the RHS.  Note: no integrals or gamma function nor use of other identities without proving them nor generating functions. No Time in Top Panel (Gnome Classic) I've been using Ubuntu with Gnome Classic for a while and I've noticed recently that in the "Indicator Applet Complete" in the Top Panel, my name and the clock have disappeared.    I think it happened a while ago when I was messing around with packages. Every time I did apt-get... it would say "the following packages can be removed by apt-get autoremove", so I did it and then the clock and my name stopped appearing.   Can anyone help me? Thanks. It would be greatly appreciated. Making sense of principal component analysis, eigenvectors & eigenvalues In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman? How to solve "sudo: /etc/sudoers.d is world writable" Find a bijection from $\mathbb R \to  \mathbb R-\mathbb N$. I want to set my function up such that all natural numbers get mapped to $n+.1$ and reals of the form $n+.1$ to $2n+.1$ is this correct?