$e^x$ as a limit proof
About $\lim \left(1+\frac {x}{n}\right)^n$
Convergence of $\int_0^\infty $sin$ (x^p) dx$ I'm looking for a way to insert a &lt;style&gt; tag into an HTML page with JavaScript.  The best way I found so far:  var divNode = document.createElement("div"); divNode.innerHTML = "&lt;br&gt;&lt;style&gt;h1 { background: red; }&lt;/style&gt;"; document.body.appendChild(divNode);   This works in Firefox, Opera and Internet Explorer but not in Google Chrome. Also it's a bit ugly with the &lt;br&gt; in front for IE.  Does anyone know of a way to create a &lt;style&gt; tag that   Is nicer Works with Chrome?   Or maybe   This is a non-standard thing I should avoid Three working browsers are great and who uses Chrome anyway? Picture rotation Is there a way to quickly rotate a picture in the SE editor without saving it on disk and rotating it in some image tool and reuploading it?  E.g. consider the picture below, assume I would like to quickly rotate it 90 degrees and wanted to do it expressly in the editor without the drudgery of saving it on disk, is there a shortcut in the stack exchange &quot;markup language&quot;?  If this feature is not available, I would like to nominate it for implementation. Nested sequence of non-empty compact subsets - intersection differs from empty set I have the following proposition:  "If $(S_n)$ is a sequence of compact subsets of a metric space such that  $S_1 \supseteq S_2 \supseteq ... \supseteq S_n$  Then $\bigcap^\infty_{i=1} S_i$ $\neq$ $\emptyset$"  My lecture notes provide the following proof, whose last step I don't understand:    Proof: Assume by way of contradiction that $\bigcap^\infty_{i=1} S_i$ = $\emptyset$. Let $V_n := (S_n)^c$ for each $n$. The class $\mathbb{C} :=\{V_n: n \in \mathbb{N} \}$ is an open covering of $S_1$. Since $S_1$ is compact, $\exists I$ finite, $I \subset \mathbb{N}$, s.t $\{V_i: i \in I \}$ covers $S_1$. Let $n:=max(I)$. The class $\{V_1, ..., V_n \}$ also covers $S_1$ and we get the following contradiction   $\emptyset = \bigcap^n_{i=1} S_i = S_n$    So, I don't get the part from "Let $n:=max(I)$..." onwards, especially the contradiction. I have tried applying DeMorgan's Law and the only way the last equalities makes sense is if the union of $\{V_1, ..., V_n \}$ equals the whole metric space, a property I haven't been able to understand from the steps of the proof. This is a corollary to the finite intersection property, so it must have a connection but again I can't quite get it.  Thanks in advance Query a region within a larger region with GeoDjango REST API I have a working Django REST API which can query all the counties in a country. //My model class kenya_counties(models.Model):   objectid = models.BigIntegerField()   id_field = models.BigIntegerField()   county_nam = models.CharField(max_length=80)   const_code = models.BigIntegerField()   constituen = models.CharField(max_length=80)   county_cod = models.BigIntegerField()   shape_leng = models.FloatField()   shape_area = models.FloatField()   geom = models.MultiPolygonField(srid=4326)    def __str__(self):       return self.county_nam  //My Serializer class CountiesSerializer(serializers.ModelSerializer):   class Meta:      model = kenya_counties      fields = ('objectid', 'id_field', 'county_nam', 'const_code', 'constituen', 'county_nam', 'county_cod', 'shape_leng', 'shape_area')  //My view @api_view(['GET', 'POST']) def counties(request):    if request.method == 'GET':        data = kenya_counties.objects.all()         serializer = CountiesSerializer(data, context={'request': request}, many=True)         return Response(serializer.data)     elif request.method == 'POST':        serializer = CountiesSerializer(data=request.data)        if serializer.is_valid():            serializer.save()            return Response(status=status.HTTP_201_CREATED)                 return Response(serializer.errors, status=status.HTTP_400_BAD_REQUEST)  //My url     re_path(r'^api/gis/counties/$', views.counties),  I get . How can I query just one county and display it on a react-leaflet map as shown below? &lt;MapContainer center={position} zoom={13} scrollWheelZoom={false}&gt;     &lt;TileLayer       attribution='&amp;copy; &lt;a href=&quot;http://osm.org/copyright&quot;&gt;OpenStreetMap&lt;/a&gt; contributors'       url=&quot;https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png&quot;     /&gt; &lt;/MapContainer&gt; Understanding Euclid's proof that the number of primes is infinite. In Euclid's proof, if $p_1, p_2, \dots, p_n$ are the only primes then $p_1 \times p_2 \times \dots \times p_n + 1$ is not divisible by any of $p_1, p_2, \dots, p_n$ (because of some algebraic facts), which makes another prime and is a contradiction.   The proof makes sense logically, and I tried some numerical examples to "feel" the proof better but...  $2 \times 3 \times 5\times 7\times 11\times 13+1$ is not a prime! $2 \times 3 \times 5\times 7\times 11\times 13 \times 17+1$ is also not prime! Why is the general case proof is not working for these examples? When using find, how do I return the file name and the line number when searching for a string? I manage to return the file name in one command and the line numbers with another one, but I can't seem to combine them.  File names: find . -type f -exec grep -l 'string to search' {} \;  Line numbers: find . -type f -exec grep -n 'string to search' {} \; SQL Server: Extract Table Meta-Data (description, fields and their data types) I'm a European citizen living in the US on a work visa. Due to some unfortunate circumstances, I need to urgently visit my home country. However I would like to get back to the US after my visit is complete. Flying from the US to Europe is not a problem as there are still daily flights between some city pairs. However going back could be tricky, as  anyone who has been in the Schengen area in the past 14 days from re-entering the country.   Is there a flight route that would let me from Europe to some country, stay there for 14 days and then fly to the US? Canada is out, as they've banned international visitors. Mexico seems to keep their border open, but I  any direct European flights from their airports. Almost every other country either suspended flights or doesn't allow foreign nationals to enter. remove oldest files I`m trying to delete old files from directory and leave only 3 newest files.   cd /home/user1/test  while [ `ls -lAR | grep ^- | wc -l` &lt; 3 ] ; do      rm `ls -t1 /home/user/test | tail -1`     echo " - - - "  done   something is wrong with conditional statement. What do the symbols displayed by ls -F mean? I noticed that if I run ls -F on a directory, some of the entries have a * or a @ after them.   spuder@ubuntu:~$ ls -F /sbin acpi_available*   getpcaps*           lvmconf*                 ntfscp*        start-stop-daemon* agetty*           getty*              lvmdiskscan@             ntfslabel*     status@ alsa*             halt@               lvmdump*                 ntfsresize*    stop@ alsactl*          hdparm*             lvmsadc@      spuder@ubuntu:~$ ls -F ~ daq-0.6.1/  examples.desktop       noname-cache.lib  snort-2.9.1/   Templates/ Desktop/    jpgraph-1.27.1/        noname.sch        snortfiles/    Ubuntu One/ Documents/     According to the ls man pages   spuder@ubuntu:~$ man ls ... -F, --classify   append indicator (one of */=&gt;@|) to entries ...   I'm guessing that @ means symbolic link,     What do these other indicators mean ( */=&gt;@| )  ? \includegraphics: Dots in filename \includegraphics cannot handle filenames that contain more than the one dot, separating the filename from the extension. Apparently it uses everything after the first dot as extension and then, of course, complains about an unknown graphics extension.  This is annoying as I very often have filenames that contain parameter values, e.g. plot_a0.4_b0.6.pdf.  Is there a way to teach LaTeX to interpret only the string behind the last dot as extension? How can I edit this trigger to only fire once? The trigger creates a related record when a checkbox is updated. The problem is it is creating a new related record everytime the record is edited. I want to edit the trigger to only create the related record once. (When the criteria is first met). The code is below  Trigger copyEquipmentonsite on Unit_Placements_Sales__c(after insert, after update) {      List&lt;Services__c&gt; sub=new List&lt;Services__c&gt;();      for(Unit_Placements_Sales__c u : Trigger.new) {           if(u.Service_Month_is_Next_Month__c == TRUE) {                 Services__c s = new Services__c();                 s.Equipment_Onsite__c=u.ID;                 s.Equipment_Type__c=u.Equipment__c;                 s.Machine_Specific_Notes__c=u.Machine_Specific_Notes__c;                    s.Product__c=u.The_Product__c;                   s.Dell_Tag_Number__c=u.Dell_Tag_Number__c;                 s.Serial_Number__c=u.Serial_Number__c;                  s.XD_Number__c=u.XD_Number__c;                 sub.add(s);           }           if(sub.size()&gt;0)           insert sub;      } } Proof: Show there is set of $n+1$ points in $\mathbb{R}^n$ such that distance between any two distinct points is $1$? Graphing y=x^2 and y=x SQL Query; Admin Log System; Foreign Keys; Retriving multiple rows form different tables at once Table A (users)  id     | name   | ip 52433    Yashas   1.1.1.1 512      Admin    5.6.7.8   Table B (admin_record)  id    | admin | time    | date    | reason | action  52433   512     2312352   1213213   Test     WARN   id is FOREIGN key  Table B keeps a log of all actions an administrator takes on his users. And the table A stores the account information of the users &amp; administrators.  I am trying to display a list of actions taken by a administrators on a user.   I want to write a query which gets me all the rows of table B where id = fixed number.  Here is what I want to display:  "Administrator1 [Name]" took action on "User Name" for "[Note]" on XX/XX/XXXX. "Administrator2 [Name]" took action on "User Name" for "[Note]" on XX/XX/XXXX. "Administrator3 [Name]" took action on "User Name" for "[Note]" on XX/XX/XXXX.   All I have is the user's id using which I need to obtain all the entries in the log along with the administrator names. I need to obtain the admin's name, users name and all the contents of table B related to a particular user in one single query. If possible, I want the to change the field name(not literally change, but want the SQL server to send the data with new field names to separate admin name &amp; user name) of the admin's name to admin_name and user's name as . I mean something like this  SELECT users.name AS admin_name WHERE users.id = %i  AND SELECT users.name AS user_name WHERE users.id = %i   (the query isn't correct,it us just an example to show what I am trying)  Progress so far:   SELECT player.name, player.ip, admin.name, record.date, record.time, record.action, record.reason FROM admin_record AS record JOIN users AS player ON player.id = record.id JOIN users AS admin ON admin.id = record.admin WHERE player.id=%i   where admin_record is table B &amp; users is table A. Unfair 3 Sided Die Question Say you have an unfair 3-sided die.  Side 1 will be rolled with probability $P_1.$ Side 2 will be rolled with probability $P_2.$ Side 3 will be rolled with probability $P_3.$  Find a formula for the probability that the 8th 3 will be rolled on the 10th roll.  Here is how I started:  $$ \sum_{i=8}^{10}P(T_8 = i) $$  Im just having trouble with that expansion? Proving the universal mapping property for polynomial rings How can I convert the binary string $x_bin="0001001100101" to its numeric value $x_num=613 in Perl? Clarification on Clausius Inequality Consider a heat reservoir which gains heat $Q$ irreversibly at temperature $T$ from the surroundings which is at temperature $T_0$. The entropy change of reservoir is then given by $\frac{Q}{T}$, while that of the surroundings is $-\frac{Q}{T_0}$.  My question is, how is this possible? According to the Clausius inequality, the entropy change of a irreversible process is greater than that due to heat transfer. Please help, thank you!