Solving a question by mathematical induction
Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
GeoJSON does not appear on my Leaflet map I'm a beginner in Leaflet and I need help to deal with GeoJSON files.  I succeeded in creating a Leaflet basemap. I can add markers without troubles but I cannot manage to show a GeoJSON layer.  Here is the code I picked up from Leaflet Tutorials website  :  var geojsonFeature = {     "type": "Feature",     "properties": {         "name": "Coors Field",         "amenity": "Baseball Stadium",         "popupContent": "This is where the Rockies play!"     },     "geometry": {         "type": "Point",         "coordinates": [-104.99404, 39.75621]     } }; L.geoJSON(geojsonFeature).addTo(map);   Unfortunatly, the GeoJSON does not appear in my map.   Can you tell me where I missed something ?    Alright, you're fine Francisco, this part was OK. I changed the whole script and it works !  However, would you know how to load a GeoJSON from an external source ?  I tried this :   var geojsonLayer = new L.GeoJSON.AJAX("foo.geojson");        geojsonLayer.addTo(map);   from   but this definitely does not work.  Any ideas ?    I am not skilled enough to understand your advice.  Here is the whole code I am using to create my map :  &lt;!DOCTYPE html&gt; &lt;html&gt; &lt;head&gt; &lt;title&gt;data&lt;/title&gt; &lt;meta charset="UTF-8"&gt; &lt;link rel="stylesheet" href="https://unpkg.com/leaflet@1.0.3/dist/leaflet.css" /&gt; &lt;script src="https://unpkg.com/leaflet@1.0.3/dist/leaflet.js"&gt;&lt;/script&gt; &lt;script src="https://code.jquery.com/jquery-3.1.1.min.js"&gt;&lt;/script&gt;   &lt;script type="text/javascript" src="http://maps.stamen.com/js/tile.stamen.js?v1.3.0"&gt;&lt;/script&gt; &lt;script src="/js/leaflet-0.7.2/leaflet.ajax.min.js"&gt;&lt;/script&gt; &lt;script src="htdocs/webmap101/leaflet.ajax.min.js" type="text/javascript" &gt;&lt;/script&gt; &lt;style&gt;    #map {width: 960px; height:700px;} &lt;/style&gt; &lt;/head&gt; &lt;body&gt;   &lt;div id="map"&gt;&lt;/div&gt; &lt;script&gt;   // create map var lat = 44.933331; var lon = 4.9;  var map = L.map('map',{center: [lat,lon],zoom: 6});   // basemaps var base = L.tileLayer('http://{s}.tile.osm.org/{z}/{x}/{y}.png',   {attribution: '© &lt;a href="http://osm.org/copyright"&gt;OpenStreetMap&lt;/a&gt; contributors'  }); var toner = new L.StamenTileLayer("toner"); var watercolor = new L.StamenTileLayer("watercolor"); var landscape = L.tileLayer('https://tile.thunderforest.com/landscape/{z}/{x}/{y}.png'); base.addTo(map);   // markers  var mark1=L.marker([lat,lon]) mark1.addTo(map)    // layers panel  L.control.layers({"base": base, "water": watercolor, "toner": toner , "landscape" : landscape}).addTo(map);    // geojson files var geojsonFeature = {     "type": "Feature",     "properties": {         "name": "Coors Field",         "amenity": "Baseball Stadium",         "popupContent": "This is where the Rockies play!"     },     "geometry": {         "type": "Point",         "coordinates": [-104.99404, 39.75621]     } }; L.geoJSON(geojsonFeature).addTo(map);    //ajax  var geojsonLayer = new L.GeoJSON.AJAX("foo.geojson");        geojsonLayer.addTo(map);  &lt;/script&gt; &lt;/body&gt; &lt;/html&gt;   I know it's a bit messy but this is actually my very first HTML script... I guess I did not include in a proper way the AJAX library.    Actually, I just want a solution to replace the lines where I create the GeoJSON by something that tells javascript to load the GeoJSON stored locally in my folder. This :  var geojsonFeature = {     "type": "Feature",     "properties": {         "name": "Coors Field",         "amenity": "Baseball Stadium",         "popupContent": "This is where the Rockies play!"     },     "geometry": {         "type": "Point",         "coordinates": [-104.99404, 39.75621]     } };   should be transformed into something like (I don't know if it exists) :  var geojsonFeature = [foo.geosjon]   where "foo.geosjson" is a file like :  { "type": "FeatureCollection", "name": "abies", "crs": { "type": "name", "properties": { "name": "urn:ogc:def:crs:OGC:1.3:CRS84" } }, "features": [ { "type": "Feature" Suppose $f$ is entire and    $$\iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$Prove that $f\equiv 0.$   So far I have:  Suppose $f$ is bounded.  Then $f$ is constant by virtue of Liouville and so the conclusion is obvious.  Thus, assume $f$ non-bounded.  Then  $$\Big|\iint_\mathbb{C}f^2(z)dA\Big| \leq \iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$ We can parameterize \begin{align*} \iint_\mathbb{C}f^2(z)dA &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;d\theta\;rdr \\ &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;ire^{i\theta}[-i\frac{1}{r}e^{-i\theta}]d\theta\;rdr \\ &amp;= \int_0^{\infty}\oint_{C_R}f^2(z)\;[-i\frac{1}{z}]dz\;rdr \\ &amp;= \int_0^{\infty}2\pi r dr\frac{1}{2\pi i}\oint_{C_R}\frac{f^2(z)}{z}dz \\ \\ &amp;= 2\pi f^2(0)\int_0^\infty rdr  \end{align*}  Whence f(0) = 0.  And I have no idea where to go from there.  The "entire" bit seems to suggest Liouville but I have already dealt with the bounded case.  Please advise ... Why are there not generic phone OS installers? How to hide TabPage from TabControl Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$ Show that for any two events $A$ and $B$, $$(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$$   Using the Inclusion-Exclusion formula, De Morgan's law and $P(AB^c)=P(A)-P(AB)$, I was able to expand the $\rm{l.h.s.}$ as $$(x+y)(x+y-4z)+2z(2z+1)+x(x-2)+y(y-2)+1\quad...\quad(*),$$ where $x=P(A),y=P(B)$ and $z=P(AB)$.  However I'm not being able to group the terms involved in the $\rm{l.h.s.}$ to form a 'suitable' expression and probably the way I have grouped them is not correct.  The expression $(*)$ is subject to the condition   $\max\{0,x+y-1\}\le z\le \min\{x,y\}$ and obviously $0\le x,y,z\le 1$, but I am unable to use this.      Is there an elegant/slick way to show the inequality? It probably can be shown without a lot of work and it's possible I'm missing something trivial here. My wireless/WiFi connection does not work. What information is needed to diagnose the issue? Your situation   You have successfully installed Ubuntu.  You have just downloaded and booted Ubuntu live media.  The latest LTS (see also ) or latest non-LTS release are preferred. See the list of  that are currently supported.) You upgraded your Ubuntu installation to the latest release that the software updater offered you. WiFi worked before, but not now on the new release. You  to new hardware.   Your problem   The wireless of your laptop or desktop is not working. You tried switching the wireless switch off and on and you tried rebooting several times, but you don't see any WiFi access points. You can see your wireless access point, but you cannot establish a connection. You want to analyze the problem, but you don't know where to start or what information you can provide.   This question and answer was written to give people the wireless script that will provide the information needed in most cases to diagnose their wireless issue's.   Related questions Is a match forfeited if a player refuses to leave after being sent off? Finding determinant of matrix without expanding HD bad blocks: is there a way to repair them? Stop setInterval call in JavaScript I am using setInterval(fname, 10000); to call a function every 10 seconds in JavaScript. Is it possible to stop calling it on some event?   I want the user to be able to stop the repeated refresh of data. Can I use Named Credential merge fields as URL parameters on the endpoint? I have a named credential as follows:  Name   Test_Credential URL    https://www.somesite.com Identity Type    Named Principal Authentication Protocol    Password Authentication Username   foobar Password   mySomeSiteDeveloperKey Generate Authorization Header   FALSE Allow Merge Fields in HTTP Header    TRUE Allow Merge Fields in HTTP Body     TRUE   The service that I am trying to authenticate against uses URL parameters to pass in the developer key and username. Is there any way to use  to pass those in to the URL?   The Named Credential attributes seem not to get decoded until the request is actually sent, as debugging the endpoint, or headers if I set them per the linked doc, show the un-merged data. I can get the callout to go to the right place, and return the expected error because the auth information is not provided. Anybody know of a "show me my HTTP request" site that I could point to to get a better idea of how this works?  I've tried both of these:  req.setEndpoint('callout:Test_Credential/api/action?id={!$Credential.Password}&amp;account={!$Credential.UserName}');  req.setEndpoint('callout:Test_Credential/api/action' + '?id=' + '{!$Credential.Password}' + '&amp;account=' + '{!$Credential.UserName}');   (Edited to remove the part about $Credential vs. $Test_Credential as I was able to determine that it is the former, and see the merged creds by calling out to .) How to import a .max model into Blender? I've tried to import but I don't see the (.max) file as an option when importing. Is there another way to import the 3DMax model into Blender for rigging and animation? We're trying to debug some applications performing broadcast.  What is the difference between the broadcast address 255.255.255.255 and as e.g. reported by ifconfig,  Bcast:192.168.1.255 How can I escape white space in a bash loop list? I have a bash shell script that loops through all child directories (but not files) of a certain directory.  The problem is that some of the directory names contain spaces.    Here are the contents of my test directory:  $ls -F test Baltimore/  Cherry Hill/  Edison/  New York City/  Philadelphia/  cities.txt   And the code that loops through the directories:  for f in `find test/* -type d`; do   echo $f done   Here's the output:   test/Baltimore test/Cherry Hill test/Edison  test/New York City test/Philadelphia   Cherry Hill and New York City are treated as 2 or 3 separate entries.  I tried quoting the filenames, like so:  for f in `find test/* -type d | sed -e 's/^/\"/' | sed -e 's/$/\"/'`; do   echo $f done   but to no avail.  There's got to be a simple way to do this.    The answers below are great.  But to make this more complicated - I don't always want to use the directories listed in my test directory.  Sometimes I want to pass in the directory names as command-line parameters instead.  I took Charles' suggestion of setting the IFS and came up with the following:  dirlist="${@}" (   [[ -z "$dirlist" ]] &amp;&amp; dirlist=`find test -mindepth 1 -type d` &amp;&amp; IFS=$'\n'   for d in $dirlist; do     echo $d   done )   and this works just fine unless there are spaces in the command line arguments (even if those arguments are quoted).  For example, calling the script like this: test.sh "Cherry Hill" "New York City" produces the following output:   Cherry Hill New York City One of my friends said that I would die if I drank distilled water (we were using it in a chemistry experiment) I gave it a go and surprisingly did not die.  I did a bit of  and found    It said that drinking only this kind of water could definitely cause death, as distilled water was highly hypotonic and it would make the blood cells expand and finally explode and ultimately cause death.  I wanted to know exactly how much of this water on an average was needed to be consumed to cause death. What is the distance between the centers? Two unit circles overlap in such a way that exactly half the area of each circle is within the overlap. How far apart are the centers of the two circles? Can we insert or update FieldSets using apex code? Although we can create or update fieldsets using salesforce UI but Is there any way to insert or update FieldSets using apex code (using metadata or API calls) ? When should you use struct and not class in C#? My conceptual model is that structs are used in times when the item is merely a collection of value types.  A way to logically hold them all together into a cohesive whole.  I came across these rules :   A struct should represent a single value. A struct should have a memory footprint less than 16 bytes. A struct should not be changed after creation.   Do these rules work? What does a struct mean semantically? 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Can I configure Ubuntu to install software/packages on the HDD instead of the SSD?