As a starting point I have started to use the fact, that every subset of $\mathbb{N}$ countable is. A $\subset \mathbb{N}$ is also a subset of $ \mathbb{N}$ and at this point I can't go forward. Or can I eventually use somehow Hilbert's Hotel theorem? I appreciate any help a lot.
Is there any way to solve this problem using the diagonalization method? I know there's the proof that uses $\mathcal{P}(\mathbb{N})$ = set of all finite subsets $\cup$ set of all infinite subsets, but I'm trying to solve it using the diagonalization method specifically. I was thinking that we can assume the two have the same cardinality and arrive at a contradiction, but I don't know how to get there. How can we show there has to be some infinite set in the set of all infinite subsets that cannot match 1-1 with a natural number?
What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$    I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$.   I know that the integral of $\frac{1}{n+1}$ is $\ln(n+1)$. What am I missing?  Thank you! What is the difference between DPI (dots per inch) and PPI (pixels per inch)? There are lots of other questions on Graphic Design which partly cover this, e.g. .  However, I have become frustrated at the number of questions and answers which confuse the two terms. I think understanding the difference is important.  So here's a place to answer this question well and clear up the confusion! Proof of "the continuous image of a connected set is connected" None of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question!   In Rudin Theorem 4.22, we know that      If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, and $E$ is a connected subset of $X$, then $f(E)$ is connected.    In the proof, we started with consider $f(E) = A \cup B$, where $A$ and $B$ are nonempty separated subsets. Then put $G = E \cap f^{-1}(A)$ and $H = E \cap f^{-1}(B)$. Then Rudin is claiming that $E = G \cup H$. I'm a little suspicious about this. What if $f$ is non-surjective, then $f^{-1}(A) \cup f^{-1}(B)$  is only a proper subset of $E$? Is there a property of $f$ being continuous that forces $f$ to be 1-1? Do we really need reals? It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.  Since almost all real numbers are not computable, it seems that real numbers are a set too big and most of them are unnecessary.  So why do mathematicians love this plethoric set of numbers so much? C++ handling very large integers I am using the RSA Algorithm for encryption/decryption, and in order to decrypt the files you have to deal with some pretty big values. More specifically, things like   P = C^d % n   = 62^65 % 133   Now that is really the only calculations that ill be doing. I have tried using Matt McCutchen's BigInteger Library, but I am getting a lot of compiler errors during linking, such as:  encryption.o(.text+0x187):encryption.cpp: undefined reference to `BigInteger::BigInteger(int)'  encryption.o(.text+0x302):encryption.cpp: undefined reference to `operator&lt;&lt;(std::ostream&amp;, BigInteger const&amp;)'  encryption.o(.text$_ZNK10BigIntegermlERKS_[BigInteger::operator*(BigInteger const&amp;) const]+0x63):encryption.cpp: undefined reference to `BigInteger::multiply(BigInteger const&amp;, BigInteger const&amp;)'   So I was wondering what would be the best way to go about handling the really big integers that come out of the RSA Algorithm.  I heard that a possibility would be to declare your variables as a double long, so...  long long decryptedCharacter;   but I'm not sure exactly how big of an integer that can store.    Well for example, I try to compile and run the following program using dev C++:  #include iostream  #include "bigint\BigIntegerLibrary.hh"  using namespace std;  int main() {     BigInteger a = 65536;     cout &lt;&lt; (a * a * a * a * a * a * a * a);     return 0; }   then I get those errors.  Derek, I thought that by including the BigIntegerLibrary.hh file, that the compiler would go through and compile all the necessary files that it will use.  How should I try and compile the program above in order to resolve the linking errors? Display header and footer in all PDF pages using VF My vf page     &lt;apex:page renderAs="pdf" Controller="testpdf1"&gt;     &lt;head&gt;         &lt;style type="text/css" media="print"&gt;             @page              {                 size: 8.5in 11in;/* width height */             }              @page              {                 @top-center                  {                     content: element(header);                 }                  @bottom-left                  {                     content: element(footer);                 }             }              div.header              {                 padding: 10px;                 position: running(header);             }              div.footer              {                 display: block;                 padding: 5px;                 position: running(footer);             }              .pagenumber:before              {                 content: counter(page);             }              .pagecount:before              {                 content: counter(pages);             }         &lt;/style&gt;     &lt;/head&gt;      &lt;div class="header"&gt;             &lt;div align="right"&gt;Header&lt;/div&gt;     &lt;/div&gt;      &lt;div class="footer"&gt;         &lt;div&gt;Page &lt;span class="pagenumber"/&gt; of &lt;span class="pagecount"/&gt;&lt;/div&gt;     &lt;/div&gt;      &lt;div class="content"&gt;         &lt;p&gt;             &lt;apex:repeat value="{!opp}" var="item"&gt;                 &lt;tr&gt;                     &lt;td class="tableContent"&gt;{!item.Name}&lt;/td&gt;                 &lt;/tr&gt;             &lt;/apex:repeat&gt;         &lt;/p&gt;     &lt;/div&gt; &lt;/apex:page&gt;   Controller  public class testpdf1 {     public list&lt;Account&gt; opp{get;set;}      public testpdf1()     {         opp=[SELECT Name  from Account limit 50];     } }   I am getting all records in pdf format but able to get header in first page only and i am not getting header from second pages how can i display header to all pages. Gravity in 2d space and inverse linear law In our three-dimensional universe, gravity obeys the inverse square law. In a four-dimensional universe, gravity would be expected to obey the inverse cube law et cetera.  In a two-dimensional universe, one would similarly expect gravity to obey an inverse linear law. I've seen it claimed that this is not actually true, when you work it out according to general relativity there would actually be no gravity at all in such a universe. Is this true? If so, is there a layman's explanation of why? Is it related to the fact that gravitational potential wells would be infinitely deep?  Would electrostatic forces follow the same law as gravity?  If there existed infinitely long strings in our universe, would they have the same gravitational effect as point particles in a 2d universe? Ubuntu 14.04 System Logging Didn't Harry have any other family he could have searched for? Adding PDF (especially GeoPDF) to ArcMap as data? I receive a lot of PDF maps (vector, image, with georeference and without) and I am having to convert them to an image file before I can add them to ArcMap and georeference.    I can't see any obvious out-of-the-box utility that would allow me to add and georeference a PDF directly without having to convert to an image file first.   Any ideas?  I would prefer to add a PDF to map directly without any prior "pdf to image" conversion How are one-time pads distributed? It seems that OTP encryption is unbreakable. I have read some articles (Wikipedia, Blogpost, etc.), but I could not find any information about how they are used.  If I wanted to create a messenger service, how would I distribute the OTP keys? Both the sender and the receiver need this key before the message is sent, right?   And how do I make sure no one else gets these keys? $\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$. I tried to assume that there is an isomorphism and to get a contradiction.. didn't succeed Why do the less than symbol () appear wrong as upside down exclamation (¡) or question mark (¿)? When typing &lt; or &gt; in LaTeX and compiling with pdflatex, the less than and greater than symbols appear at upside down exclamation points. I'm not in math mode. Custom Post Type Archives by Year & Month? How do you display the archives of a Custom Post Type by Year &amp; Month? Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$ JavaScript variables declare outside or inside loop? What's the difference between term_id and term_taxonomy_id IllegalArgumentException: Invalid column Planet where humans fight for access to algae Set out on another planet where a human spaceship lands for a research and they observer a huge mass moving below them. On closer look this mass appears to be humans, and the density of the population increases towards the sea. The reason for this is algae,the only nutrition found on the planet. On a beach they discover tall, huge humans (developed due to the natural selection), controlling the best access to the algae. Eventually the researchers die, because humans eat them. Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$