I have to prove: $||x|-|y||\leq |x+y|\leq |x|+|y|$ I have already proved $|x+y|\leq |x|+|y|$.  I saw proofs for $||x|-|y||\leq |x-y|$, can I use the proof for  $||x|-|y||\leq |x-y|$ and just add that $|x-y|\leq|x+y|$?
I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much.
Why do we use microwaves in microwave oven? We know that any object above absolute zero emits electromagnetic radiation. So hotter the object shorter the wavelengths. In the electromagnetic radiation spectrum radio waves has the longest then microwave, then infrared, then visible light, then ultraviolet, then x-ray, then gamma rays. Why do we use microwaves in microwave oven when infrared and visible light are much hotter and how do microwaves cook food when they are cooler than visible light and others. How do I convert a Word file's references to .bib format How do I convert a Word file's references to .bib format? Is any tool available  to convert it to .bib format? How to print only defined variables (shell and/or environment variables) in bash The bash builtin command set, if invoked without arguments, will print all shell and environment variables, but also all defined functions. This makes the output unusable for humans and difficult to grep.  How can I make the bash builtin command set print only the variables and not the functions?  Are there other commands which prints only the shell variables, without the functions?  Note: bash differentiates between shell variables and environment variables. see here Primary decomposition of a monomial ideal Disable commandButton after first click to prevent double submission I have an apex:commandButton on a visualforce page that invokes a method on the controller. There is currently no rerender value set for the button. Instead the controller method returns a PageReference to redirect the user as required.  E.g.  Visualforce:  &lt;apex:commandButton value="Save" action="{!save}"/&gt;   Controller:  public PageReference save() {     // save body ...      // return the user the a parent opportunity     return new PageReference('/' + opp.Id); }   Users are currently able to click on the resulting save button multiple times before the controller completes and returns the PageReference to redirect the browser to an opportunity.  How can I disable the commandButton after the first click?    I tried wrapping the commandButton in an actionStatus and facet, but the need to define the rerender property prevented the resulting PageReference redirect.  E.g.  This will render nicely in the browser and prevent multiple clicks, but doesn't redirect to the Opportunity on completion.  &lt;apex:actionStatus id="saveStatus"&gt;     &lt;apex:facet name="stop"&gt;         &lt;apex:commandButton value="Save" action="{!save}" status="saveStatus" rerender="saveParentBlock" /&gt;     &lt;/apex:facet&gt;     &lt;apex:facet name="start"&gt;         &lt;apex:commandButton value="Saving..." disabled="true" status="saveStatus"/&gt;     &lt;/apex:facet&gt; &lt;/apex:actionStatus&gt;     There is a similar question . I need to support an existing PageReference redirect, which slightly alters the requirements.  Force.com Discussion Boards:   Ideas: Converting Polygon into single or all Points using QGIS? Transferring Fallout 4 Saves (PS4) Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? Zero probability and impossibility I read a comment under :     There are plenty of events that can occur that have zero probability.   This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering    if zero probability and impossibility mean the same? if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?   Thanks and regards! Algorithm to detect intersection of two rectangles? I'm looking for an algorithm to detect if two rectangles intersect (one at an arbitrary angle, the other with only vertical/horizontal lines).  Testing if a corner of one is in the other ALMOST works.  It fails if the rectangles form a cross-like shape.  It seems like a good idea to avoid using slopes of the lines, which would require special cases for vertical lines. Can I travel to the USA while working remotely for my non-US employer? Ubuntu 18.04 bond with netplan duplicate MAC address after cloning Select longest continuous sequence I am trying to construct a query in PostgreSQL 9.0 that gets the longest sequence of continuous rows for a specific column.  Consider the following table:  lap_id (serial), lap_no (int), car_type (enum), race_id (int FK)   Where lap_no is unique for each (race_id, car_type).  I would like the query to produce the longest sequence for a given race_id and car_type, so it would return an int (or long) that is highest.  With the following data:  1, 1, red, 1 2, 2, red, 1 3, 3, red, 1 4, 4, red, 1 5, 1, blue, 1 6, 5, red, 1 7, 2, blue, 1 8, 1, green, 1   For car_type = red and race_id = 1 the query would return 5 as the longest sequence of the lap_no field.  I found a similar question  however my situation is a bit more straightforward.  (I would also like to know the longest sequence for a given car_type for all races, but was planning to work that out myself.) The search parameter isanswered ignores acceptances How to render world position pass If a graph with $n$ vertices and $n$ edges there must a cycle? About $\lim \left(1+\frac {x}{n}\right)^n$ Can spells with movable areas of effect be moved out of sight or behind obstacles? With spells like Moonbeam or Flaming Sphere, how should we treat the movement of their areas-of-effect when it comes to passing them behind a thick pillar or around a blind corner?  Should we treat any movement of them as if they are being 're-cast' (and so requiring a clear unobstructed view)? Or could a mage presumably will them somewhere out-of-sight? I have a vocabulary like this:   Parent 1  Child 1 Child 2 Child 3   Parent 2  Child 1 Child 2    Now I would like to use a view to create a list of a Parent term with its Child terms. I try to do this with Views UI, but only show the term parent... No the child term's. It is very confusing for me.  Can anyone help with this case? Thanks :D failed to get D-Bus connection: Operation not permitted I'm trying to list services on my CentOS image running in Docker using  systemctl list-units     but I get this error message:  Failed to get D-Bus connection: Operation not permitted   Any suggestions what the problem might be?