This is a Lemma from N.L. Carothers Real Analysis.  Lemma. An infinite subset $A$ of $\mathbb{N}$ is countable.  Proof. Since $A\ne\emptyset$, there is a smallest element $x_1\in A$. Then $A\backslash\{x_1\}\ne\emptyset$, and there must be a smallest $x_2\in A\backslash\{x_1\}$. By induction we can find $x_1,x_2,\dotsc,x_n,\dotsc\in A$, where $x_n=\min(A\backslash\{x_1,\dotsc,x_{n-1}\})$. Now, suppose that $x\in A\backslash\{x_1,x_2,\dotsc\}\ne\emptyset$. Then the set $\{k:x_k&gt;x\}$ must be nonempty, and hence it has a least element. That is, there is some $n$ with $x_1&lt;x_2&lt;\cdots&lt;x_{n-1}&lt;x&lt;x_n$. But this contradicts the choice of $x_n$.  The part of the proof that I don't understand is that the set $\{k:x_k&gt;x\}$ must be nonempty. Carothers claims that otherwise we would have $x\in A$ and $x&lt;x_1=\min A$. I can't prove that if $\{k:x_k&gt;x\}=\emptyset$ then $x&lt;x_1$.  Help appreciated please.
In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$.
LaTeX doesn't recognize "et al." in the Bibliography I'm using Jabref to manage my Bibliography.  Sometimes, I put directly:     First Author et al.   when adding the article author's name in Jabref.  When compiling my .tex file, LaTeX doesn't recognize the et. al as more names. It takes it as a name/surname.  This is an example of a Bibliography entry in Jabref.  @ARTICLE{Wolf2003,     author = {Wolf, K.et al.},     title = {Compensation mechanism in tumor cell migration: mesenchymal–amoeboid     transition after blocking of pericellular proteolysis},     journal = {J. Cell Biol.},     year = {2003},     volume = {160},     pages = {267–277},    }   This is a .texMWE with the bib entry mentioned:  \documentclass[11pt]{book}   \usepackage{cite}   \bibliographystyle{unsrt} \begin{document}   Here is the bib entry mentioned\cite{Wolf2003}    \bibliography{bibfile}  \end{document}   Here's the output:   How can I solve this problem? 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While in Ireland, I want to apply for US business visa.  Why not to apply from Egypt? Because the waiting time is too long (75 days). The waiting time in Ireland in only 18 days. Both visits (Ireland, US) are short visits (3 weeks for each). How to secure Elliptic Curve ElGamal encryption against known plaintext attacks? If I have an encoding function  $f(x)$ that maps a message $m$ to a point $P$ on a suitable Elliptic Curve $E$ . If I have the public key $Q$ of my recepient then I can encrypt the message as follows:  Choosing a random number $k &lt; n -1 $ where $n$ is the order of the curve E .  Calculating $C = [k] G $ where $G$ is the generator point of the curve E   Calculating $R = [k]Q $ where $Q$ is the recipient's public key   Now the message is encrypted by adding $P$ to $R$ to get          $C_e$ $( = R + P)$ which is sent along with $C$ to the recepient.  DECRYPTION:  Now for the recepient to decrypt my message they compute $R = [privatekey] C $ (where $C = [k]G $) and simply subtract $R$ from $C_e$ to get the encoded point $P$ back which is then decoded by an inverse function $f^{-1}(x)$ to recover $m$.  Now , two problems arise : If the attacker knows the plaintext (suppose if I follow a particular format of data while sending messages ) he could encode his guessed plaintext and subtract it from $C_e$ to recover $ R $ back ! . If I used that same $ R  $ to encrypt further "blocks" of my message then the security of the later parts of my message has been breached!   I realize that doing  something like a scalar multiplication over and over again for every block of data, would be a huge drawback in efficiency and speed.   Is there any way to compute a different $ R $ for every block of data, without using much resources and that too quickly ?   Does reusing the same $R$ for another message break security ?  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And more interesting, why it will compile when I cast MEMBER to int?  #include &lt;vector&gt;  class Foo {   public:       static const int MEMBER = 1;   };  int main(){       vector&lt;int&gt; v;       v.push_back( Foo::MEMBER );       // undefined reference to `Foo::MEMBER'     v.push_back( (int) Foo::MEMBER ); // OK       return 0; } I am a Physics undergrad, and just started studying Topology. How do you define neighborhood and open set in Topology.Wikipedia gives a circular definition.   An  is defined as follows.      In topology, a set  is called an open set if it is a neighborhood of every point    While a  is defined as follows:     If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$, which includes an open set $U$ containing $p$   which itself contains the term open set.  How do you define it exactly? I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. 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