I need to prove that function  $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is pain. $\frac{|x-y|}{1 + |x-y|}$ + $\frac{|y-z|}{1 + |y-z|} ≥ \frac{|x-z|}{1 + |x-z|} ⇒ \frac{|x-y| + |y-z| + 2|(x-y)(y-z)|}{1 + |x-y| + |y-z| + |(x-y)(y-z)|}≥ \frac{|x-z|}{1 + |x-z|}$, but then I am stuck. Can somebody show me way out of this?
I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches.  Recently I've bought Topology by D. Kahn and am finding it difficult.  Here is a problem that I think I'm am answering sufficiently but any help would be great if I am off.  If $d$ is a metric on a set $S$, show that  $$d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is a metric on $S$.  The conditions for being a metric are $d(X,Y)\ge{0}, d(X,Y)=0$ iff $X=Y$, $d(X,Y)=d(Y,X)$, and $d(X,Y)\le{d(X,Z)+d(Z,Y)}$.  Thus, we simply go axiom by axiom.  1)  Since both $d(x,y)\ge{0}$ and $1+d(x,y)\ge{0},$ it is clear that $d_1(x,y)\ge{0}$.  (Is this a sufficient analysis?) 2) $d_1(x,x)=\frac{d(x,x)}{1+d(x,x)}=\frac{0}{1+0}=0$. 3) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}=\frac{d(y,x)}{1+d(y,x)}=d_1(y,x).$ 4) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}\le{\frac{d(x,z)+d(z,y)}{1+d(x,z)+d(z,y)}}=\frac{d(x,z)}{1+d(x,z)+d(z,y)}+\frac{d(z,y)}{1+d(x,z)+d(z,y)}\lt\frac{d(x,z)}{1+d(x,z)}+\frac{d(z,y)}{1+d(z,y)}=d_1(x,z)+d_1(z,y).$  However, #4 is strictly less, not less than or equal to, according to my analysis, so where did I go wrong?
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$. Can't boot without Flash Drive plugged in Please do not "lie" that the Community user "modified" the question If the Community user bumps a question, It says It has modified it:    However, It did not really modify the question, It rather just visited it. The word "modified" is a bit unfortunate IMHO because it suggests that something changed in the question or its answers, which is not true. I therefore suggest that we look for another word which would be more descriptive and less confusing. Some come to my (and others') mind:   visited, poked, bumped, promoted, dredged up, brought up, ...   (Actually, "Bumped" is even officially used now:  and .)  I'm not a native speaker and maybe someone find a better word. To me, visited sounds the best, but that may be only me. It makes sense, too, since Community really just visits the questions, with the site-effect of them getting bumped. It shouldn't confuse anybody.  Actually, it can be harmful in some sense, since , and it is interesting to know that this has happened, so that more people can check that the edit makes sense. You therefore can't simply ignore all "edited by Community" messages. (Thanks to Monica Cellio for pointing it out.)  (Remark: I know this has been questioned before (    ) but I think it has never been requested to change the action title to something more descriptive. If I'm wrong, I apologize.) (The "lying" it the question title is meant as a sarcasm) How do you generate all the permutations of a list in Python, independently of the type of elements in that list?  For example:  permutations([]) []  permutations([1]) [1]  permutations([1, 2]) [1, 2] [2, 1]  permutations([1, 2, 3]) [1, 2, 3] [1, 3, 2] [2, 1, 3] [2, 3, 1] [3, 1, 2] [3, 2, 1] Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests?  Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on  seems to agree:     Molecular and quantum physics show that the electromagnetic force is the fundamental interaction responsible for contact forces.    An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what force is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such.  I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual "source needed" objections, and that this topic is either misunderstood by or poorly covered in the standard texts.  (The argument can be found ).   Which way is it? How do I choose a macro lens for a Canon APS-C camera? Apply a string replacement on the citation as returned by \cite Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $ Can't join a community I look for similar questions to this one but my problem is different. I want to join this community:    so I click in     but I got this error:    but then, I found that the community is already in list of communities:    However, if I click on the community, I'm an anonymous member     and even I have the option to join again     but if I click on it, the problem starts again :( what can I do? How to get a cross-origin resource sharing (CORS) post request working Particular nodes are missing from node editor How do I select all the columns in a table that only contain NULL values for all the rows?  I'm using MS SQL Server 2005.  I'm trying to find out which columns are not used in the table so I can delete them. Correlating filename with package Say, I'm given a filename like libasound.so.2. Is there a way to query the package manager to correlate the filename with its corresponding package?  I'm interested to get answers for any package manager, but my primary focus is on apt and yum. Prove that $\frac{(n-2)s^2}{\sigma^2}\sim \chi^{2}_{n-2}$ Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. So I am running on Windows 7 (64-bit) Home Premium with 12 GB RAM (Intel Core i7 920).  I have restricted the Page File to 400MB since it was eating up a lot of space om my SSD (that is 80GB).   After that I sometimes get the "low memory"-warning, like this, except for the behold-comment:   When I get those warning I have usually a lot of RAM available; when I check Resource Manager I have over 2GB "Free" and over 2GB "Available" - usually more than that.  The diskspace on all my drives have over 10GB free.  So the question is - why does Windows complain? It actually restarted by computer (hard boot) when it happened, and as I said - lots or RAM available. Installing Steam on Debian I have ran into some issues when it went down to install Steam on Debian. The console as well as Apper comes up with missing dependencies  and the package isn't going to install. Counterspelling a counterspell Are +1/+1 counters different from +1/+1 effects? Is a +1/+1 counter I get from, say Unleash or Undying different than +1/+1 from something like  or ?