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Proving that the triangle inequality holds for a metric on $\mathbb{C}$ Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as:   $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} +  \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$   I have done the proof of the first two propositions for being a metric, but I'm having a problem in proving the triangle inequality. What are the benefits/detriments to graduate students joining unions? Show that $A$ is an invertible matrix if $\left(A+I\right)^3=0$ and find $A^{-1}$ Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$  I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine. "modified" link in the home page questions list does not contain "?lastactivity" parameter I have just noticed that the "modified" link for  does not contain "?lastactivity" parameter in homepage questions list:        When , it's changing to https://meta.stackexchange.com/questions/253359/list-of-communities-with-base-css-updates-completed/286876#286876 as it should, and showing the deleted answer to those who can see it.  Other questions I checked are not affected. Possible way to reproduce might be question with zero answers, where the "modified" was caused by a now-deleted answer. I need to add 30 minutes to values in a Oracle date column.  I do this in my SELECT statement by specifying   to_char(date_and_time + (.000694 * 31)  which works fine most of the time.  But not when the time is on the AM/PM border.  For example, adding 30 minutes to 12:30 [which is  PM] returns 1:00 which is AM. The answer I expect is 13:00.  What's the correct way to do this? What is the best way to add options to a select from a JavaScript object with jQuery? What is the best method for adding options to a &lt;select&gt; from a JavaScript object using jQuery?  I'm looking for something that I don't need a plugin to do, but I would also be interested in the plugins that are out there.  This is what I did:  selectValues = { "1": "test 1", "2": "test 2" };  for (key in selectValues) {   if (typeof (selectValues[key] == 'string') {     $('#mySelect').append('&lt;option value="' + key + '"&gt;' + selectValues[key] + '&lt;/option&gt;');   } }   A clean/simple solution:  This is a cleaned up and simplified :  $.each(selectValues, function(key, value) {      $('#mySelect')           .append($('&lt;option&gt;', { value : key })           .text(value)); });   Changes from matdumsa's: (1) removed the close tag for the option inside append() and (2) moved the properties/attributes into an map as the second parameter of append(). Show that if $n&gt;2$, then $(n!)^2&gt;n^n$.     My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n&gt;(n+1)^{n-1}$ Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get anything. Please help! Why does light in a room not form constructive and destructive interference patterns? Can black holes actually merge? If time stops at the event horizon, can we ever detect two black holes merging? In other words, if you are a short distance away, would you encounter a spherically symmetric gravitational field, or a dipole field? How to pass parameters to an alias? How to Count Duplicates in List with LINQ $$\int^2_{-2}\frac{x^2}{1+5^x}$$  How do I start to integrate this?  I know the basics and tried substituting $5^x$ by $u$ where by changing the base of logarithm I get $\frac{\ln(u)}{\ln 5}=x$, but I got stuck.  Any hints would suffice preferably in the original question and not after my substitution.  (And also using the basic definite integrals property.)  Now I know only basic integration, that is restricted to high school, so would prefer answer in terms of that level. When I visit  I see this:     The "Congratulations! You've earned (badge)" section has dropped down completely beneath the tabs under it. The "You've earned a new privilege!" section has a gap at the bottom:     Shog9 says in comments how to reproduce this:     Gotta have earned a badge, not dismissed the notification, and be looking at your own profile. Javascript close alert box I want to be able to close an alert box automatically using Javascript after a certain amount of time or on a specific event (i.e. onkeypress).  From my research, it doesn't look like that's possible with the built-in alert() function.  Is there a way to override it and have control over the dialog box that it opens?  Also, I don't want an override that shows a hidden div as the alert.  I need an actual dialog box. Deprecated: Function split() is deprecated. How to rewrite this statement? Is there a C++ decompiler? All of my Firefox add-ons have been disabled suddenly, how can I re-enable them? All of my Firefox extensions were spontaneously and simultaneously disabled (converted to Legacy status) a few moments ago.  How can I re-enable them? Firefox 66.0.3 (64-bit) Homomorphism between compact Hausdorff spaces. Can I answer my own questions, even if I knew the answer before asking? I might want to answer my own question because:  I arrived at a solution before there were any other correct answers or I like mine better, can I still answer my own question?  I have a folder with lists of code snippets for things I always forget or get wrong, such as regular expressions for processing XML, common WinForms control idioms, etc. Is it considered poor etiquette for Stack Overflow to ask a question I already know the answer to, and then self answer, just so it gets recorded on the website?   Related:          For more information, see &quot;&quot; in the .